[ { "id": "Probability_0000", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "financial mathematics", "probability" ], "problem_v1": "Suppose that a roulette wheel is spun. What is the probability that a number between 12 and 27 (inclusive) comes up? Answer=[ANS]", "answer_v1": [ "0.421052631578947" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that a roulette wheel is spun. What is the probability that a number between 7 and 31 (inclusive) comes up? Answer=[ANS]", "answer_v2": [ "0.657894736842105" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that a roulette wheel is spun. What is the probability that a number between 9 and 27 (inclusive) comes up? Answer=[ANS]", "answer_v3": [ "0.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0001", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "financial mathematics", "probability" ], "problem_v1": "Suppose that a single die with 23 sides (numbered 1, 2, 3,..., 23) is rolled once. What is the probability of getting an even number? Answer=[ANS]", "answer_v1": [ "0.478260869565217" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that a single die with 15 sides (numbered 1, 2, 3,..., 15) is rolled once. What is the probability of getting an even number? Answer=[ANS]", "answer_v2": [ "0.466666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that a single die with 17 sides (numbered 1, 2, 3,..., 17) is rolled once. What is the probability of getting an even number? Answer=[ANS]", "answer_v3": [ "0.470588235294118" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0002", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "Counting" ], "problem_v1": "A computer retail store has $14$ personal computers in stock. A buyer wants to purchase $3$ of them. Unknown to either the retail store or the buyer, $3$ of the computers in stock have defective hard drives. Assume that the computers are selected at random.\n(a) In how many different ways can the $3$ computers be chosen? answer: [ANS]\n(b) What is the probability that exactly one of the computers will be defective? answer: [ANS]\n(c) What is the probability that at least one of the computers selected is defective? answer: [ANS]", "answer_v1": [ "364", "0.453296703296703", "0.546703296703297" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A computer retail store has $8$ personal computers in stock. A buyer wants to purchase $4$ of them. Unknown to either the retail store or the buyer, $4$ of the computers in stock have defective hard drives. Assume that the computers are selected at random.\n(a) In how many different ways can the $4$ computers be chosen? answer: [ANS]\n(b) What is the probability that exactly one of the computers will be defective? answer: [ANS]\n(c) What is the probability that at least one of the computers selected is defective? answer: [ANS]", "answer_v2": [ "70", "0.228571428571429", "0.985714285714286" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A computer retail store has $10$ personal computers in stock. A buyer wants to purchase $3$ of them. Unknown to either the retail store or the buyer, $3$ of the computers in stock have defective hard drives. Assume that the computers are selected at random.\n(a) In how many different ways can the $3$ computers be chosen? answer: [ANS]\n(b) What is the probability that exactly one of the computers will be defective? answer: [ANS]\n(c) What is the probability that at least one of the computers selected is defective? answer: [ANS]", "answer_v3": [ "120", "0.525", "0.708333333333333" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0003", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "1", "keywords": [ "Counting" ], "problem_v1": "Determine the size of the sample space that corresponds to the experiment of tossing a coin the following number of times:\n(a) $4$ times answer: [ANS]\n(b) $7$ times answer: [ANS]\n(c) $n$ times answer: [ANS]", "answer_v1": [ "16", "128", "2^n" ], "answer_type_v1": [ "NV", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Determine the size of the sample space that corresponds to the experiment of tossing a coin the following number of times:\n(a) $2$ times answer: [ANS]\n(b) $9$ times answer: [ANS]\n(c) $n$ times answer: [ANS]", "answer_v2": [ "4", "512", "2^n" ], "answer_type_v2": [ "NV", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Determine the size of the sample space that corresponds to the experiment of tossing a coin the following number of times:\n(a) $2$ times answer: [ANS]\n(b) $7$ times answer: [ANS]\n(c) $n$ times answer: [ANS]", "answer_v3": [ "4", "128", "2^n" ], "answer_type_v3": [ "NV", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0004", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "Random Variable", "Discrete" ], "problem_v1": "(a) Count the number of ways to arrange a sample of $5$ elements from a population of $10$ elements. NOTE: Order is not important. answer: [ANS]\n(b) If random sampling is to be employed, the probability that any particular sample will be selected is [ANS]", "answer_v1": [ "252", "0.00396825396825397" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Count the number of ways to arrange a sample of $1$ elements from a population of $12$ elements. NOTE: Order is not important. answer: [ANS]\n(b) If random sampling is to be employed, the probability that any particular sample will be selected is [ANS]", "answer_v2": [ "12", "0.0833333333333333" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Count the number of ways to arrange a sample of $2$ elements from a population of $10$ elements. NOTE: Order is not important. answer: [ANS]\n(b) If random sampling is to be employed, the probability that any particular sample will be selected is [ANS]", "answer_v3": [ "45", "0.0222222222222222" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0005", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "3", "keywords": [ "algebra", "combination" ], "problem_v1": "A baseball player has a batting average of 0.295. What is the probability that he has exactly 6 hits in his next 7 at bats?\nThe probability is [ANS].", "answer_v1": [ "0.00325251458622398" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A baseball player has a batting average of 0.385. What is the probability that he has exactly 1 hits in his next 7 at bats?\nThe probability is [ANS].", "answer_v2": [ "0.145817438243779" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A baseball player has a batting average of 0.3. What is the probability that he has exactly 3 hits in his next 7 at bats?\nThe probability is [ANS].", "answer_v3": [ "0.2268945" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0006", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "3", "keywords": [ "algebra", "probability" ], "problem_v1": "A ball is drawn randomly from a jar that contains 8 red balls, 6 white balls, and 6 yellow ball. Find the probability of the given event.\n(a) A red ball is drawn; The probability is: [ANS]\n(b) A white ball is drawn; The probability is: [ANS]\n(c) A yellow ball is drawn; The probability is: [ANS]", "answer_v1": [ "0.4", "0.3", "0.3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A ball is drawn randomly from a jar that contains 2 red balls, 9 white balls, and 3 yellow ball. Find the probability of the given event.\n(a) A red ball is drawn; The probability is: [ANS]\n(b) A white ball is drawn; The probability is: [ANS]\n(c) A yellow ball is drawn; The probability is: [ANS]", "answer_v2": [ "0.142857142857143", "0.642857142857143", "0.214285714285714" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A ball is drawn randomly from a jar that contains 4 red balls, 6 white balls, and 4 yellow ball. Find the probability of the given event.\n(a) A red ball is drawn; The probability is: [ANS]\n(b) A white ball is drawn; The probability is: [ANS]\n(c) A yellow ball is drawn; The probability is: [ANS]", "answer_v3": [ "0.285714285714286", "0.428571428571429", "0.285714285714286" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0007", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "algebra", "combination" ], "problem_v1": "You flip a fair coin 10 times. What is the probability that it lands on heads exactly 6 times?\nThe probability of exactly 6 heads is [ANS].\nWhat is the probability that it lands on heads at least 6 times?\nThe probability of at least 6 heads is [ANS].", "answer_v1": [ "0.205078125", "0.376953125" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You flip a fair coin 10 times. What is the probability that it lands on heads exactly 3 times?\nThe probability of exactly 3 heads is [ANS].\nWhat is the probability that it lands on heads at least 3 times?\nThe probability of at least 3 heads is [ANS].", "answer_v2": [ "0.1171875", "0.9453125" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You flip a fair coin 10 times. What is the probability that it lands on heads exactly 4 times?\nThe probability of exactly 4 heads is [ANS].\nWhat is the probability that it lands on heads at least 4 times?\nThe probability of at least 4 heads is [ANS].", "answer_v3": [ "0.205078125", "0.828125" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0008", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "algebra", "probability" ], "problem_v1": "A die is rolled. Find the probability of the given event.\n(a) The number showing is a 5; The probability is: [ANS]\n(b) The number showing is an even number; The probability is: [ANS]\n(c) The number showing is greater than 3; The probability is: [ANS]", "answer_v1": [ "0.166666666666667", "0.5", "0.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A die is rolled. Find the probability of the given event.\n(a) The number showing is a 2; The probability is: [ANS]\n(b) The number showing is an even number; The probability is: [ANS]\n(c) The number showing is greater than 5; The probability is: [ANS]", "answer_v2": [ "0.166666666666667", "0.5", "0.166666666666667" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A die is rolled. Find the probability of the given event.\n(a) The number showing is a 3; The probability is: [ANS]\n(b) The number showing is an even number; The probability is: [ANS]\n(c) The number showing is greater than 4; The probability is: [ANS]", "answer_v3": [ "0.166666666666667", "0.5", "0.333333333333333" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0009", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "statistics", "probability" ], "problem_v1": "For each problem, select the best response.\n(a) A phenomenon is observed many, many times under identical conditions. The proportion of times a particular event occurs is recorded. This proportion represents [ANS] A. the probability of the event. B. the variance of the event. C. the correlation of the event. D. the distribution of the event.\n(b) The collection of all possible outcomes of a random phenomenon is called [ANS] A. a census. B. the distribution. C. the sample space. D. the probability. E. None of the above.", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) The probability of any outcome of a random phenomenon is [ANS] A. either 0 or 1, depending on whether or not the phenomenon can actually occur. B. any number, as long as it is a value between 0 and 1. C. the precise degree of randomness present in the phenomenon. D. the proportion of a very long series of repetitions in which the outcome occurs.\n(b) The collection of all possible outcomes of a random phenomenon is called [ANS] A. a census. B. the distribution. C. the probability. D. the sample space. E. None of the above.", "answer_v2": [ "D", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) The probability of any outcome of a random phenomenon is [ANS] A. either 0 or 1, depending on whether or not the phenomenon can actually occur. B. the proportion of a very long series of repetitions in which the outcome occurs. C. any number, as long as it is a value between 0 and 1. D. the precise degree of randomness present in the phenomenon.\n(b) The collection of all possible outcomes of a random phenomenon is called [ANS] A. a census. B. the sample space. C. the probability. D. the distribution. E. None of the above.", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0010", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "1", "keywords": [ "statistics", "concepts", "introduction", "binomial distribution" ], "problem_v1": "One die is rolled. List the outcomes comprising the following events: (make sure you use the correct notation with the set braces {}, put comma between the outcomes and do not put space between them)\n(a) event the die comes up odd answer: [ANS]\n(b) event the die comes up 4 or more answer: [ANS]\n(c) event the die comes up at most 2 answer: [ANS]", "answer_v1": [ "(1,3,5)", "(4,5,6)", "(1,2)" ], "answer_type_v1": [ "UOL", "UOL", "UOL" ], "options_v1": [ [], [], [] ], "problem_v2": "One die is rolled. List the outcomes comprising the following events: (make sure you use the correct notation with the set braces {}, put comma between the outcomes and do not put space between them)\n(a) event the die comes up 3 answer: [ANS]\n(b) event the die comes up odd answer: [ANS]\n(c) event the die comes up even answer: [ANS]", "answer_v2": [ "(3)", "(1,3,5)", "(2,4,6)" ], "answer_type_v2": [ "UOL", "UOL", "UOL" ], "options_v2": [ [], [], [] ], "problem_v3": "One die is rolled. List the outcomes comprising the following events: (make sure you use the correct notation with the set braces {}, put comma between the outcomes and do not put space between them)\n(a) event the die comes up odd answer: [ANS]\n(b) event the die comes up 4 or more answer: [ANS]\n(c) event the die comes up 3 answer: [ANS]", "answer_v3": [ "(1,3,5)", "(4,5,6)", "(3)" ], "answer_type_v3": [ "UOL", "UOL", "UOL" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0011", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "1", "keywords": [ "statistics", "introduction", "concepts" ], "problem_v1": "Determine whether the following number can possibly be probability. Write \"YES\" for yes and \"NO\" for no. (without quotations)\n(a) 1 answer: [ANS]\n(b) 3.6 answer: [ANS]\n(c)-2.4 answer: [ANS]\n(d) 0 answer: [ANS]\n(e) $\\frac{2}{3}$ answer: [ANS]\n(f) 0.7 answer: [ANS]", "answer_v1": [ "YES", "NO", "NO", "YES", "YES", "YES" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Determine whether the following number can possibly be probability. Write \"YES\" for yes and \"NO\" for no. (without quotations)\n(a) 0 answer: [ANS]\n(b) 1 answer: [ANS]\n(c) 0.1 answer: [ANS]\n(d) 4 answer: [ANS]\n(e) $\\frac{1}{2}$ answer: [ANS]\n(f)-2.9 answer: [ANS]", "answer_v2": [ "YES", "YES", "YES", "NO", "YES", "No" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Determine whether the following number can possibly be probability. Write \"YES\" for yes and \"NO\" for no. (without quotations)\n(a) 1 answer: [ANS]\n(b) 3.6 answer: [ANS]\n(c) 0 answer: [ANS]\n(d)-2.7 answer: [ANS]\n(e) $\\frac{2}{3}$ answer: [ANS]\n(f) 0.3 answer: [ANS]", "answer_v3": [ "YES", "NO", "YES", "No", "YES", "YES" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0012", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "sample space" ], "problem_v1": "In an experiment, a ball is drawn from an urn containing 13 orange balls and 11 purple balls. If the ball is orange, two coins are tossed. Otherwise three coins are tossed. How many elements of the sample space will have a orange ball? [ANS]\nHow many elements of the sample space are there altogether? [ANS]", "answer_v1": [ "13*2^2", "11*2^3+13*2^2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "In an experiment, a ball is drawn from an urn containing 7 red balls and 14 yellow balls. If the ball is red, three coins are tossed. Otherwise two coins are tossed. How many elements of the sample space will have a red ball? [ANS]\nHow many elements of the sample space are there altogether? [ANS]", "answer_v2": [ "7*2^3", "14*2^2+7*2^3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "In an experiment, a ball is drawn from an urn containing 9 green balls and 11 orange balls. If the ball is green, two coins are tossed. Otherwise three coins are tossed. How many elements of the sample space will have a green ball? [ANS]\nHow many elements of the sample space are there altogether? [ANS]", "answer_v3": [ "9*2^2", "11*2^3+9*2^2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0013", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "probability" ], "problem_v1": "An experiment consists of choosing a subset from a fixed number of objects where the arrangement/order of the chosen objects is not important. Determine the size of the sample space when you choose the following:\n(a) 6 objects from 14 Answer: [ANS]\n(b) 6 objects from 24 Answer: [ANS]\n(c) 5 objects from 15 Answer: [ANS]", "answer_v1": [ "3003", "134596", "3003" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An experiment consists of choosing a subset from a fixed number of objects where the arrangement/order of the chosen objects is not important. Determine the size of the sample space when you choose the following:\n(a) 7 objects from 29 Answer: [ANS]\n(b) 2 objects from 13 Answer: [ANS]\n(c) 3 objects from 15 Answer: [ANS]", "answer_v2": [ "1560780", "78", "455" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An experiment consists of choosing a subset from a fixed number of objects where the arrangement/order of the chosen objects is not important. Determine the size of the sample space when you choose the following:\n(a) 8 objects from 20 Answer: [ANS]\n(b) 5 objects from 12 Answer: [ANS]\n(c) 3 objects from 15 Answer: [ANS]", "answer_v3": [ "125970", "792", "455" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0014", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "1", "keywords": [ "Probability", "sample space", "complement" ], "problem_v1": "French is the official language of the Democratic Republic of Congo. Suppose that in the adult population of the capital city of the country, Kinshasa, \\% can speak French. Suppose two adults are chosen at random from the population of Kinshasa. Let A denote the event that both adults selected can speak French. What is the complement of event A? [ANS] A. B. $\\left\\{\\begin{array}{c} (\\text{Speaks French}, \\, \\text{Does not speak French}) \\\\ (\\text{Does not speak French}, \\, \\text{Speaks French}) \\\\ (\\text{Does not speak French}, \\, \\text{Does not speak French}) \\end{array} \\right\\}$ C. $\\begin{array}{c} \\left\\{(\\text{Does not speak French}, \\, \\text{Speaks French}) \\right\\} \\\\ \\end{array}$ D. $\\begin{array}{c} \\left\\{(\\text{Does not speak French}, \\, \\text{Does not speak French}) \\right\\} \\end{array}$ E.", "answer_v1": [ "B" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "French is the official language of the Democratic Republic of Congo. Suppose that in the adult population of the capital city of the country, Kinshasa, \\% can speak French. Suppose two adults are chosen at random from the population of Kinshasa. Let A denote the event that both adults selected can speak French. What is the complement of event A? [ANS] A. $\\begin{array}{c} \\left\\{(\\text{Speaks French}, \\, \\text{Does not speak French}) \\right\\} \\\\ \\end{array}$ B. C. $\\left\\{\\begin{array}{c} (\\text{Does not speak French}, \\, \\text{Does not speak French}) \\\\ (\\text{Speaks French}, \\, \\text{Does not speak French}) \\\\ (\\text{Does not speak French}, \\, \\text{Speaks French}) \\end{array} \\right\\}$ D. E. $\\begin{array}{c} \\left\\{(\\text{Does not speak French}, \\, \\text{Does not speak French}) \\right\\} \\end{array}$", "answer_v2": [ "C" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "French is the official language of the Democratic Republic of Congo. Suppose that in the adult population of the capital city of the country, Kinshasa, \\% can speak French. Suppose two adults are chosen at random from the population of Kinshasa. Let A denote the event that both adults selected can speak French. What is the complement of event A? [ANS] A. B. $\\begin{array}{c} \\left\\{(\\text{Speaks French}, \\, \\text{Does not speak French}) \\right\\} \\\\ \\end{array}$ C. D. $\\left\\{\\begin{array}{c} (\\text{Does not speak French}, \\, \\text{Does not speak French}) \\\\ (\\text{Speaks French}, \\, \\text{Does not speak French}) \\\\ (\\text{Does not speak French}, \\, \\text{Speaks French}) \\end{array} \\right\\}$ E. $\\begin{array}{c} \\left\\{(\\text{Does not speak French}, \\, \\text{Does not speak French}) \\right\\} \\end{array}$", "answer_v3": [ "D" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0015", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "3", "keywords": [ "Sets", "Venn diagrams", "complements", "unions", "intersections of events", "laws for sets", "identifying subsets" ], "problem_v1": "The players on a soccer team wear shirts, with each player having one of the numbers 1, 2,..., 11 on their backs. The set A contains players with even numbers on their shirts. The set B comprises players wearing an odd number less than 7. The set C contains the defenders, which are those wearing numbers less than 6. Select the correct set that corresponds to each of the following.\nPart a) $(A \\cap B^c) \\cup (B \\cap C)^c$ [ANS] A. $\\{2,4,6,7,8,9,10,11\\}$ B. $\\{6,7,8,10,11\\}$ C. $\\{6,7,8,9,11\\}$ D. $\\{2,3,4,5,6,8,10\\}$ E. $\\{1,2,3,4,5,6,8,10\\}$\nPart b) $(A \\cap B) \\cup (A \\cap C)$ [ANS] A. $\\{2,4\\}$ B. $\\{1,2,3,4,5\\}$ C. $\\emptyset$ D. $\\{1,3,5\\}$ E. $\\{2\\}$", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The players on a soccer team wear shirts, with each player having one of the numbers 1, 2,..., 11 on their backs. The set A contains players with even numbers on their shirts. The set B comprises players wearing an odd number less than 7. The set C contains the defenders, which are those wearing numbers less than 6. Select the correct set that corresponds to each of the following.\nPart a) $A \\cap (B \\cup C)$ [ANS] A. $\\{2\\}$ B. $\\emptyset$ C. $\\{2,4\\}$ D. $\\{1,2,3,4,5\\}$ E. $\\{1,3,5\\}$\nPart b) $(A \\cap B^c) \\cup (B \\cap C)^c$ [ANS] A. $\\{2,4,6,7,8,9,10,11\\}$ B. $\\{6,7,8,10,11\\}$ C. $\\{2,3,4,5,6,8,10\\}$ D. $\\{1,2,3,4,5,6,8,10\\}$ E. $\\{6,7,8,9,11\\}$", "answer_v2": [ "C", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The players on a soccer team wear shirts, with each player having one of the numbers 1, 2,..., 11 on their backs. The set A contains players with even numbers on their shirts. The set B comprises players wearing an odd number less than 7. The set C contains the defenders, which are those wearing numbers less than 6. Select the correct set that corresponds to each of the following.\nPart a) $(A \\cap B) \\cup (A \\cap C)$ [ANS] A. $\\{1,3,5\\}$ B. $\\{2\\}$ C. $\\emptyset$ D. $\\{2,4\\}$ E. $\\{1,2,3,4,5\\}$\nPart b) $(A^c \\cup B^c) \\cap C^c$ [ANS] A. $\\{6,7,8,9,10,11\\}$ B. $\\{1,2,3,4,5\\}$ C. $\\{1,3,5\\}$ D. $\\{5,7,9,11\\}$ E. $\\{6,7,8,9,11\\}$", "answer_v3": [ "D", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0016", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "limits", "factoring", "rational function" ], "problem_v1": "When tossing a coin and rolling a die at the same time, the number of outcomes in the sample space is [ANS] A. 12 B. 2 C. 8 D. 6 E. 24", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "When tossing a coin twice, the number of outcomes in the sample space is [ANS] A. 2 B. 6 C. 3 D. 4 E. 5", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "When tossing a coin twice, the number of outcomes in the sample space is [ANS] A. 5 B. 2 C. 3 D. 6 E. 4", "answer_v3": [ "E" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0017", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "1", "keywords": [ "probability", "subjective", "marginal", "conditional" ], "problem_v1": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. Probability refers to a number between 0 and 1, which expresses the chance that an event will occur. [ANS] 2. Marginal probability is the probability that a given event will occur, with no other events taken into consideration. [ANS] 3. Two or more events are said to be independent when the occurrence of one event has no effect on the probability that the other will occur. [ANS] 4. A useful graphical method of constructing the sample space for an experiment is pie chart.", "answer_v1": [ "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. Marginal probability is the probability that a given event will occur, with no other events taken into consideration. [ANS] 2. A useful graphical method of constructing the sample space for an experiment is pie chart. [ANS] 3. Probability refers to a number between 0 and 1, which expresses the chance that an event will occur. [ANS] 4. The annual estimate of the number of deaths of infants is an example of the classical approach to probability.", "answer_v2": [ "T", "F", "T", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. The outcome of a game of roulette based on historical data is not an example of the relative frequency approach to probability. [ANS] 2. The annual estimate of the number of deaths of infants is an example of the classical approach to probability. [ANS] 3. Probability refers to a number between 0 and 1, which expresses the chance that an event will occur. [ANS] 4. Conditional probability is the probability that an event will occur, given that another event could also occur.", "answer_v3": [ "F", "F", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Probability_0018", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "probability" ], "problem_v1": "Which of the following is a requirement of the probabilities assigned to the outcomes $O_i$: [ANS] A. $0 \\leq P(O_i)\\leq 1$ for each i B. $P(O_i)=1+P(O_i^C)$ C. $P(O_i) \\geq 1$ D. $P(O_i) \\leq 0$\nWhich of the following statements is correct given that the events A and B have nonzero probabilities? [ANS] A. A and B are always mutually exclusive B. A and B can be both independent and mutually exclusive C. A and B cannot be both independent and mutually exclusive D. A and B are always independent", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Which of the following is a requirement of the probabilities assigned to the outcomes $O_i$: [ANS] A. $P(O_i) \\geq 1$ B. $P(O_i)=1+P(O_i^C)$ C. $P(O_i) \\leq 0$ D. $0 \\leq P(O_i)\\leq 1$ for each i\nIf you roll an unbiased die 50 times, you should expect an even number to appear: [ANS] A. on the average, 25 out of the 50 rolls B. 25 out of the 50 rolls C. at least twice in the 50 rolls D. on every other roll", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Which of the following is a requirement of the probabilities assigned to the outcomes $O_i$: [ANS] A. $P(O_i) \\geq 1$ B. $0 \\leq P(O_i)\\leq 1$ for each i C. $P(O_i)=1+P(O_i^C)$ D. $P(O_i) \\leq 0$\nOf the last 500 customers entering a supermarket, 50 have purchased a loaf of bread. If the relative frequency approach for assigning probabilities is used, the probability that the next customer will purchase a loaf of bread is: [ANS] A. 0.90 B. 0.10 C. 0.50 D. None of the above answers is correct", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0019", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "probability" ], "problem_v1": "A survey asks adults to report their marital status. Suppose that in the city which the survey is conducted, 52\\% of adults are married, 13\\% are single, 23\\% are divorced, and 12\\% are widowed. Find the probabilities of each of the following events: The adult is single=[ANS]\nThe adult is not divorced=[ANS]\nThe adult is either widowed or divorced=[ANS]", "answer_v1": [ "0.13", "0.77", "0.35" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A survey asks adults to report their marital status. Suppose that in the city which the survey is conducted, 41\\% of adults are married, 15\\% are single, 20\\% are divorced, and 24\\% are widowed. Find the probabilities of each of the following events: The adult is single=[ANS]\nThe adult is not divorced=[ANS]\nThe adult is either widowed or divorced=[ANS]", "answer_v2": [ "0.15", "0.8", "0.44" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A survey asks adults to report their marital status. Suppose that in the city which the survey is conducted, 45\\% of adults are married, 13\\% are single, 21\\% are divorced, and 21\\% are widowed. Find the probabilities of each of the following events: The adult is single=[ANS]\nThe adult is not divorced=[ANS]\nThe adult is either widowed or divorced=[ANS]", "answer_v3": [ "0.13", "0.79", "0.42" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0020", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "statistics", "probability", "classical", "relative frequency", "subjective" ], "problem_v1": "There are three approaches to determining the probability that an outcome will occur: Classical, Relative Frequency, and Subjective. Which is most appropriate in determining the probability of the following outcomes? (Select from the pull-down menus.)\n[ANS] 1. A randomly selected man will suffer from athlete's foot in the coming year. [ANS] 2. The total rolled on a pair of 6-sided dice will be 9. [ANS] 3. A randomly selected woman is over six feet tall. [ANS] 4. A randomly selected share of stock will increase in price over the next month.", "answer_v1": [ "R", "C", "R", "S" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "C", "R", "S" ], [ "C", "R", "S" ], [ "C", "R", "S" ], [ "C", "R", "S" ] ], "problem_v2": "There are three approaches to determining the probability that an outcome will occur: Classical, Relative Frequency, and Subjective. Which is most appropriate in determining the probability of the following outcomes? (Select from the pull-down menus.)\n[ANS] 1. The total rolled on a pair of 6-sided dice will be 9. [ANS] 2. A randomly selected share of stock will increase in price over the next month. [ANS] 3. A randomly selected man will suffer from athlete's foot in the coming year. [ANS] 4. The unemployment rate will rise in the next month.", "answer_v2": [ "C", "S", "R", "S" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "C", "R", "S" ], [ "C", "R", "S" ], [ "C", "R", "S" ], [ "C", "R", "S" ] ], "problem_v3": "There are three approaches to determining the probability that an outcome will occur: Classical, Relative Frequency, and Subjective. Which is most appropriate in determining the probability of the following outcomes? (Select from the pull-down menus.)\n[ANS] 1. Five tosses of a coin will result in exactly two heads. [ANS] 2. The unemployment rate will rise in the next month. [ANS] 3. A randomly selected man will suffer from athlete's foot in the coming year. [ANS] 4. The next spin of a roulette wheel will produce a red number.", "answer_v3": [ "C", "S", "R", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "C", "R", "S" ], [ "C", "R", "S" ], [ "C", "R", "S" ], [ "C", "R", "S" ] ] }, { "id": "Probability_0021", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "probability" ], "problem_v1": "The natural remedy echinacea is reputed to boost the immune system, which will reduce flu and colds. A 6-month study was undertaken to determine whether the remedy works. From this study, the following probability distribution of the number of respiratory infections per year (X) for echinacea users was produced:\n$\\begin{array}{cccccc}\\hline X & 0 & 1 & 2 & 3 & 4 \\\\ \\hline P(X) & 0.332 & 0.328 & 0.205 & 0.076 & 0.059 \\\\ \\hline \\end{array}$\nFind the following probabilities: A. An echinacea user has more than one infection per year [ANS]\nB. An echinacea user has no infections per year [ANS]\nC. An echinacea user has between one and three (inclusive) infections per year [ANS]", "answer_v1": [ "0.34", "0.332", "0.609" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The natural remedy echinacea is reputed to boost the immune system, which will reduce flu and colds. A 6-month study was undertaken to determine whether the remedy works. From this study, the following probability distribution of the number of respiratory infections per year (X) for echinacea users was produced:\n$\\begin{array}{cccccc}\\hline X & 0 & 1 & 2 & 3 & 4 \\\\ \\hline P(X) & 0.342 & 0.32 & 0.209 & 0.071 & 0.058 \\\\ \\hline \\end{array}$\nFind the following probabilities: A. An echinacea user has more than one infection per year [ANS]\nB. An echinacea user has no infections per year [ANS]\nC. An echinacea user has between one and three (inclusive) infections per year [ANS]", "answer_v2": [ "0.338", "0.342", "0.6" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The natural remedy echinacea is reputed to boost the immune system, which will reduce flu and colds. A 6-month study was undertaken to determine whether the remedy works. From this study, the following probability distribution of the number of respiratory infections per year (X) for echinacea users was produced:\n$\\begin{array}{cccccc}\\hline X & 0 & 1 & 2 & 3 & 4 \\\\ \\hline P(X) & 0.34 & 0.323 & 0.206 & 0.072 & 0.059 \\\\ \\hline \\end{array}$\nFind the following probabilities: A. An echinacea user has more than one infection per year [ANS]\nB. An echinacea user has no infections per year [ANS]\nC. An echinacea user has between one and three (inclusive) infections per year [ANS]", "answer_v3": [ "0.337", "0.34", "0.601" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0022", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "probability", "random variables" ], "problem_v1": "Using historical records, the personnel manager of a plant has determined the probability of $X$, the number of employees absent per day. It is\n$\\begin{array}{ccccccccc}\\hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline P(X) & 0.0048 & 0.0246 & 0.3096 & 0.3397 & 0.2193 & 0.0793 & 0.0186 & 0.0041 \\\\ \\hline \\end{array}$\nFind the following probabilities. A. $P(2 \\leq X \\leq 5)$ Probability=[ANS]\nB. $P(X > 5)$ Probability=[ANS]\nC. $P(X < 4)$ Probability=[ANS]", "answer_v1": [ "0.9479", "0.0227", "0.6787" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Using historical records, the personnel manager of a plant has determined the probability of $X$, the number of employees absent per day. It is\n$\\begin{array}{ccccccccc}\\hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline P(X) & 0.004 & 0.025 & 0.3091 & 0.3393 & 0.22 & 0.0793 & 0.0182 & 0.0051 \\\\ \\hline \\end{array}$\nFind the following probabilities. A. $P(2 \\leq X \\leq 5)$ Probability=[ANS]\nB. $P(X > 5)$ Probability=[ANS]\nC. $P(X < 4)$ Probability=[ANS]", "answer_v2": [ "0.9477", "0.0233", "0.6774" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Using historical records, the personnel manager of a plant has determined the probability of $X$, the number of employees absent per day. It is\n$\\begin{array}{ccccccccc}\\hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline P(X) & 0.0043 & 0.0246 & 0.3093 & 0.3396 & 0.2192 & 0.0793 & 0.0188 & 0.0049 \\\\ \\hline \\end{array}$\nFind the following probabilities. A. $P(2 \\leq X \\leq 5)$ Probability=[ANS]\nB. $P(X > 5)$ Probability=[ANS]\nC. $P(X < 4)$ Probability=[ANS]", "answer_v3": [ "0.9474", "0.0237", "0.6778" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0023", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "probability" ], "problem_v1": "A survey of Amazon.com shoppers reveals the following probability distribution of the number of books per hit:\n$\\begin{array}{ccccccccc}\\hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline P(X) & 0.324 & 0.204 & 0.075 & 0.13 & 0.028 & 0.022 & 0.01 & 0.207 \\\\ \\hline \\end{array}$\nFind the following probabilities: A. What is the probability that an Amazon.com visitor will buy four books? Probability=[ANS]\nB. What is the probability that an Amazon.com visitor will buy eight books? Probability=[ANS]\nC. What is the probability that an Amazon.com visitor will not buy any books? Probability=[ANS]\nD. What is the probability that an Amazon.com visitor will buy at least one book? Probability=[ANS]", "answer_v1": [ "0.028", "0", "0.324", "0.676" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A survey of Amazon.com shoppers reveals the following probability distribution of the number of books per hit:\n$\\begin{array}{ccccccccc}\\hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline P(X) & 0.32 & 0.209 & 0.071 & 0.239 & 0.029 & 0.021 & 0.009 & 0.102 \\\\ \\hline \\end{array}$\nFind the following probabilities: A. What is the probability that an Amazon.com visitor will buy four books? Probability=[ANS]\nB. What is the probability that an Amazon.com visitor will buy eight books? Probability=[ANS]\nC. What is the probability that an Amazon.com visitor will not buy any books? Probability=[ANS]\nD. What is the probability that an Amazon.com visitor will buy at least one book? Probability=[ANS]", "answer_v2": [ "0.029", "0", "0.32", "0.68" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A survey of Amazon.com shoppers reveals the following probability distribution of the number of books per hit:\n$\\begin{array}{ccccccccc}\\hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline P(X) & 0.33 & 0.208 & 0.072 & 0.215 & 0.028 & 0.02 & 0.01 & 0.117 \\\\ \\hline \\end{array}$\nFind the following probabilities: A. What is the probability that an Amazon.com visitor will buy four books? Probability=[ANS]\nB. What is the probability that an Amazon.com visitor will buy eight books? Probability=[ANS]\nC. What is the probability that an Amazon.com visitor will not buy any books? Probability=[ANS]\nD. What is the probability that an Amazon.com visitor will buy at least one book? Probability=[ANS]", "answer_v3": [ "0.028", "0", "0.33", "0.67" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0024", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "prealgebra", "common core", "probability" ], "problem_v1": "A student is chosen at random from a class with 19 girls and 18 boys.\nThe probability of choosing a boy is [ANS]. The odds in favor of choosing a boy is [ANS]\nThe probability of choosing either a girl or a boy is [ANS]\nThe odds that neither a girl nor a boy is chosen is [ANS]", "answer_v1": [ "18/37", "18/19", "1", "0/1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A student is chosen at random from a class with 25 girls and 10 boys.\nThe probability of choosing a boy is [ANS]. The odds in favor of choosing a boy is [ANS]\nThe probability of choosing either a girl or a boy is [ANS]\nThe odds that neither a girl nor a boy is chosen is [ANS]", "answer_v2": [ "10/35", "10/25", "1", "0/1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A student is chosen at random from a class with 19 girls and 12 boys.\nThe probability of choosing a boy is [ANS]. The odds in favor of choosing a boy is [ANS]\nThe probability of choosing either a girl or a boy is [ANS]\nThe odds that neither a girl nor a boy is chosen is [ANS]", "answer_v3": [ "12/31", "12/19", "1", "0/1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0025", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "prealgebra", "common core", "probability" ], "problem_v1": "Suppose you select a letter at random from the word MISSISSIPPI.\nThe probability of selecting the letter P is [ANS]\nThe probability of selecting the letter I is [ANS]\nThe probability of selecting the letters S or M is [ANS]\nThe probability of not selecting the letter M is [ANS]", "answer_v1": [ "2/11", "4/11", "4/11+1/11", "1-1/11" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose you select a letter at random from the word MISSISSIPPI.\nThe probability of selecting the letter M is [ANS]\nThe probability of selecting the letter P is [ANS]\nThe probability of selecting the letters I or S is [ANS]\nThe probability of not selecting the letter S is [ANS]", "answer_v2": [ "1/11", "2/11", "4/11+4/11", "1-4/11" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose you select a letter at random from the word MISSISSIPPI.\nThe probability of selecting the letter I is [ANS]\nThe probability of selecting the letter S is [ANS]\nThe probability of selecting the letters M or P is [ANS]\nThe probability of not selecting the letter P is [ANS]", "answer_v3": [ "4/11", "4/11", "1/11+2/11", "1-2/11" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0026", "subject": "Probability", "topic": "Sample Space", "subtopic": "Outcomes & events", "level": "2", "keywords": [ "prealgebra", "common core", "probability" ], "problem_v1": "A fun size bag of M $\\&$ Ms has about 19 candies. You open one of the bags and discover: 4 Blues, 3 Yellows, 6 Browns, 4 Reds and 2 Greens. 4 Blues, 3 Yellows, 6 Browns, 4 Reds and 2 Greens.\nThe probability of choosing a brown is [ANS]. The odds in favor of choosing a yellow is [ANS]\nThe probability of choosing either a blue or a red is [ANS]\nThe odds against a green being chosen is [ANS]", "answer_v1": [ "6/19", "3/16", "8/19", "17/2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A fun size bag of M $\\&$ Ms has about 18 candies. You open one of the bags and discover: 2 Blues, 4 Yellows, 5 Browns, 3 Reds and 4 Greens. 2 Blues, 4 Yellows, 5 Browns, 3 Reds and 4 Greens.\nThe probability of choosing a brown is [ANS]. The odds in favor of choosing a yellow is [ANS]\nThe probability of choosing either a blue or a red is [ANS]\nThe odds against a green being chosen is [ANS]", "answer_v2": [ "5/18", "4/14", "5/18", "14/4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A fun size bag of M $\\&$ Ms has about 15 candies. You open one of the bags and discover: 2 Blues, 3 Yellows, 5 Browns, 3 Reds and 2 Greens. 2 Blues, 3 Yellows, 5 Browns, 3 Reds and 2 Greens.\nThe probability of choosing a brown is [ANS]. The odds in favor of choosing a yellow is [ANS]\nThe probability of choosing either a blue or a red is [ANS]\nThe odds against a green being chosen is [ANS]", "answer_v3": [ "5/15", "3/12", "5/15", "13/2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0027", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "1", "keywords": [ "financial mathematics", "probability" ], "problem_v1": "Suppose that $X$ is an event, and that $P(X)=0.7$. What is $P(\\bar{X})$ (i.e. the probability that $X$ will not occur)? Answer=[ANS]", "answer_v1": [ "0.3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $X$ is an event, and that $P(X)=0.16$. What is $P(\\bar{X})$ (i.e. the probability that $X$ will not occur)? Answer=[ANS]", "answer_v2": [ "0.84" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $X$ is an event, and that $P(X)=0.35$. What is $P(\\bar{X})$ (i.e. the probability that $X$ will not occur)? Answer=[ANS]", "answer_v3": [ "0.65" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0028", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "financial mathematics", "probability" ], "problem_v1": "An experiment consists of flipping a coin, rolling a 15 sided die, and spinning a roulette wheel. What is the probability that the coin comes up heads and the die comes up less than 4 and the roulette wheel comes up with a number greater than 14?\nAnswer=[ANS]", "answer_v1": [ "0.0578947368421053" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An experiment consists of flipping a coin, rolling a 9 sided die, and spinning a roulette wheel. What is the probability that the coin comes up heads and the die comes up less than 4 and the roulette wheel comes up with a number greater than 17?\nAnswer=[ANS]", "answer_v2": [ "0.0833333333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An experiment consists of flipping a coin, rolling a 11 sided die, and spinning a roulette wheel. What is the probability that the coin comes up heads and the die comes up less than 4 and the roulette wheel comes up with a number greater than 14?\nAnswer=[ANS]", "answer_v3": [ "0.0789473684210526" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0029", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "financial mathematics", "probability" ], "problem_v1": "Suppose that four 10-sided dice are tossed. What is the probability that the sum on the four dice is greater than or equal to 6? Answer=[ANS]", "answer_v1": [ "0.9995" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that five 6-sided dice are tossed. What is the probability that the sum on the five dice is greater than or equal to 7? Answer=[ANS]", "answer_v2": [ "0.999228395061728" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that four 7-sided dice are tossed. What is the probability that the sum on the four dice is greater than or equal to 6? Answer=[ANS]", "answer_v3": [ "0.997917534360683" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0030", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "Random Variable" ], "problem_v1": "Three dice are rolled. Let the random variable $x$ represent the sum of the 3 dice. By assuming that each of the $6^3$ possible outcomes is equally likely, find the probability that $x$ equals $14$. $P(x=14)=$ [ANS]", "answer_v1": [ "0.0694444444444444" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Three dice are rolled. Let the random variable $x$ represent the sum of the 3 dice. By assuming that each of the $6^3$ possible outcomes is equally likely, find the probability that $x$ equals $5$. $P(x=5)=$ [ANS]", "answer_v2": [ "0.0277777777777778" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Three dice are rolled. Let the random variable $x$ represent the sum of the 3 dice. By assuming that each of the $6^3$ possible outcomes is equally likely, find the probability that $x$ equals $8$. $P(x=8)=$ [ANS]", "answer_v3": [ "0.0972222222222222" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0031", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "Probability", "Events" ], "problem_v1": "If $P(A)=0.65,$ $P(B)=0.45$ and $P(A \\cap B)=0.15,$ find the following probabilities: a) $P(A \\cup B)=$ [ANS]\nb) $P(\\overline{A})=$ [ANS]\nc) $P(\\overline{B})=$ [ANS]\nd) $P(A \\cap \\overline{B})=$ [ANS]\ne) $P(\\overline{A \\cap B})=$ [ANS]", "answer_v1": [ "0.95", "0.35", "0.55", "0.5", "0.85" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "If $P(A)=0.55,$ $P(B)=0.6$ and $P(A \\cap B)=0.2,$ find the following probabilities: a) $P(A \\cup B)=$ [ANS]\nb) $P(\\overline{A})=$ [ANS]\nc) $P(\\overline{B})=$ [ANS]\nd) $P(A \\cap \\overline{B})=$ [ANS]\ne) $P(\\overline{A \\cap B})=$ [ANS]", "answer_v2": [ "0.95", "0.45", "0.4", "0.35", "0.8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "If $P(A)=0.55,$ $P(B)=0.55$ and $P(A \\cap B)=0.15,$ find the following probabilities: a) $P(A \\cup B)=$ [ANS]\nb) $P(\\overline{A})=$ [ANS]\nc) $P(\\overline{B})=$ [ANS]\nd) $P(A \\cap \\overline{B})=$ [ANS]\ne) $P(\\overline{A \\cap B})=$ [ANS]", "answer_v3": [ "0.95", "0.45", "0.45", "0.4", "0.85" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0032", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "Probability", "Discrete" ], "problem_v1": "The number $75$ is written as a sum of three natural numbers 75=a+b+c (the triple $(a,b,c)$ is ordered; e.g., the decompositions $75=1+1+73$ and $75=1+73+1$ are different. Also, assume that all the decompositions have equal probability.) What is the probability that there exists a triangle with sides $a$, $b$, and $c$? $\\ $ [ANS]", "answer_v1": [ "0.26027397260274" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The number $35$ is written as a sum of three natural numbers 35=a+b+c (the triple $(a,b,c)$ is ordered; e.g., the decompositions $35=1+1+33$ and $35=1+33+1$ are different. Also, assume that all the decompositions have equal probability.) What is the probability that there exists a triangle with sides $a$, $b$, and $c$? $\\ $ [ANS]", "answer_v2": [ "0.272727272727273" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The number $49$ is written as a sum of three natural numbers 49=a+b+c (the triple $(a,b,c)$ is ordered; e.g., the decompositions $49=1+1+47$ and $49=1+47+1$ are different. Also, assume that all the decompositions have equal probability.) What is the probability that there exists a triangle with sides $a$, $b$, and $c$? $\\ $ [ANS]", "answer_v3": [ "0.265957446808511" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0033", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "Probability", "Discrete" ], "problem_v1": "A quick quiz consists of $5$ multiple choice problems, each of which has $5$ answers, only one of which is correct. If you make random guesses on all $5$ problems,\n(a) What is the probability that all $5$ of your answers are incorrect? answer: [ANS]\n(b) What is the probability that all $5$ of your answers are correct? answer: [ANS]", "answer_v1": [ "0.32768", "0.00032" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A quick quiz consists of $3$ multiple choice problems, each of which has $6$ answers, only one of which is correct. If you make random guesses on all $3$ problems,\n(a) What is the probability that all $3$ of your answers are incorrect? answer: [ANS]\n(b) What is the probability that all $3$ of your answers are correct? answer: [ANS]", "answer_v2": [ "0.578703703703704", "0.00462962962962963" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A quick quiz consists of $3$ multiple choice problems, each of which has $5$ answers, only one of which is correct. If you make random guesses on all $3$ problems,\n(a) What is the probability that all $3$ of your answers are incorrect? answer: [ANS]\n(b) What is the probability that all $3$ of your answers are correct? answer: [ANS]", "answer_v3": [ "0.512", "0.008" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0034", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "Probability", "Discrete" ], "problem_v1": "An instructor gives his class a set of 18 problems with the information that the next quiz will consist of a random selection of 7 of them. If a student has figured out how to do 14 of the problems, what is the probability the he or she will answer correctly\n(a) all 7 problems? [ANS]\n(b) at least 6 problems? [ANS]", "answer_v1": [ "0.107843137254902", "0.485294117647059" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An instructor gives his class a set of 10 problems with the information that the next quiz will consist of a random selection of 5 of them. If a student has figured out how to do 6 of the problems, what is the probability the he or she will answer correctly\n(a) all 5 problems? [ANS]\n(b) at least 4 problems? [ANS]", "answer_v2": [ "0.0238095238095238", "0.261904761904762" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An instructor gives his class a set of 13 problems with the information that the next quiz will consist of a random selection of 5 of them. If a student has figured out how to do 7 of the problems, what is the probability the he or she will answer correctly\n(a) all 5 problems? [ANS]\n(b) at least 4 problems? [ANS]", "answer_v3": [ "0.0163170163170163", "0.179487179487179" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0035", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "Probability", "Discrete" ], "problem_v1": "Two fair dice are tossed, and the up face on each die is recorded. Find the probability of observing each of the following events: $A: \\{$ The sum of the numbers is equal to 5 $\\}$ $B: \\{$ The sum of the numbers is even $\\}$ $C: \\{$ A 5 appears on exactly one of the dice $\\}$ $P(A)=$ [ANS] $\\ \\ \\ \\ \\ P(B)=$ [ANS] $\\ \\ \\ \\ \\ P(C)=$ [ANS]", "answer_v1": [ "0.111111111111111", "0.5", "0.277777777777778" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Two fair dice are tossed, and the up face on each die is recorded. Find the probability of observing each of the following events: $A: \\{$ The sum of the numbers is odd $\\}$ $B: \\{$ The sum of the numbers is equal to 7 $\\}$ $C: \\{$ A 1 appears on at least one of the dice $\\}$ $P(A)=$ [ANS] $\\ \\ \\ \\ \\ P(B)=$ [ANS] $\\ \\ \\ \\ \\ P(C)=$ [ANS]", "answer_v2": [ "0.5", "0.166666666666667", "0.305555555555556" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Two fair dice are tossed, and the up face on each die is recorded. Find the probability of observing each of the following events: $A: \\{$ The difference of the numbers is 3 $\\}$ $B: \\{$ The sum of the numbers is equal to 8 $\\}$ $C: \\{$ A 2 appears on exactly one of the dice $\\}$ $P(A)=$ [ANS] $\\ \\ \\ \\ \\ P(B)=$ [ANS] $\\ \\ \\ \\ \\ P(C)=$ [ANS]", "answer_v3": [ "0.166666666666667", "0.138888888888889", "0.277777777777778" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0036", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "Probability", "Discrete" ], "problem_v1": "An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 97 students in the school. There are 36 in the Spanish class, 33 in the French class, and 23 in the German class. There are 15 students that in both Spanish and French, 7 are in both Spanish and German, and 9 are in both French and German. In addition, there are 4 students taking all 3 classes. If one student is chosen randomly, what is the probability that he or she is taking exactly one language class? [ANS]\nIf two students are chosen randomly, what is the probability that both of them are taking French? [ANS]", "answer_v1": [ "0.43298969072165", "0.11340206185567" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 92 students in the school. There are 42 in the Spanish class, 32 in the French class, and 15 in the German class. There are 15 students that in both Spanish and French, 4 are in both Spanish and German, and 6 are in both French and German. In addition, there are 2 students taking all 3 classes. If one student is chosen randomly, what is the probability that he or she is taking at least one language class? [ANS]\nIf two students are chosen randomly, what is the probability that at least one of them is taking a language class? [ANS]", "answer_v2": [ "0.717391304347826", "0.922360248447205" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 103 students in the school. There are 32 in the Spanish class, 31 in the French class, and 23 in the German class. There are 13 students that in both Spanish and French, 4 are in both Spanish and German, and 7 are in both French and German. In addition, there are 2 students taking all 3 classes. If one student is chosen randomly, what is the probability that he or she is taking at least two language classes? [ANS]\nIf two students are chosen randomly, what is the probability that neither of them is taking a language class? [ANS]", "answer_v3": [ "0.194174757281553", "0.141062250142776" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0037", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "Probability", "Discrete", "Counting" ], "problem_v1": "What is the probability that a positive integer $m$ in the range $1 \\leq m \\leq 100$, which is selected randomly, is divisible by 8? [ANS]", "answer_v1": [ "0.12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the probability that a positive integer $m$ in the range $1 \\leq m \\leq 100$, which is selected randomly, is divisible by 3? [ANS]", "answer_v2": [ "0.33" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the probability that a positive integer $m$ in the range $1 \\leq m \\leq 100$, which is selected randomly, is divisible by 5? [ANS]", "answer_v3": [ "0.2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0038", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "Probability", "Discrete" ], "problem_v1": "How many people have to be in a room in order that the probability that at least two of them celebrate their birthday on the same day is at least 0.16? (Ignore leap years, and assume that all outcomes are equally likely.) [ANS]", "answer_v1": [ "12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "How many people have to be in a room in order that the probability that at least two of them celebrate their birthday on the same day is at least 0.04? (Ignore leap years, and assume that all outcomes are equally likely.) [ANS]", "answer_v2": [ "6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "How many people have to be in a room in order that the probability that at least two of them celebrate their birthday on the same day is at least 0.08? (Ignore leap years, and assume that all outcomes are equally likely.) [ANS]", "answer_v3": [ "9" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0039", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "4", "keywords": [ "Probability", "Discrete" ], "problem_v1": "A group of kids containing 18 boys and 16 girls is lined up in random order-that is, each of the 34! permutations is assumed to be equally likely. What is the probability that the person in the 14-th position is a boy? [ANS]", "answer_v1": [ "0.529411764705882" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A group of kids containing 10 boys and 20 girls is lined up in random order-that is, each of the 30! permutations is assumed to be equally likely. What is the probability that the person in the 6-th position is a girl? [ANS]", "answer_v2": [ "0.666666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A group of kids containing 13 boys and 16 girls is lined up in random order-that is, each of the 29! permutations is assumed to be equally likely. What is the probability that the person in the 8-th position is a boy? [ANS]", "answer_v3": [ "0.448275862068966" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0040", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "Probability", "Discrete" ], "problem_v1": "A financial firm is performing an assessment test and relies on a random sampling of their accounts. Suppose this firm has $6582$ customer accounts numbered from $0001$ to $6582$. One account is to be chosen at random. What is the probability that the selected account number is $3258$? answer: [ANS]", "answer_v1": [ "0.000151929504709815" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A financial firm is performing an assessment test and relies on a random sampling of their accounts. Suppose this firm has $6932$ customer accounts numbered from $0001$ to $6932$. One account is to be chosen at random. What is the probability that the selected account number is $1249$? answer: [ANS]", "answer_v2": [ "0.000144258511252164" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A financial firm is performing an assessment test and relies on a random sampling of their accounts. Suppose this firm has $6606$ customer accounts numbered from $0001$ to $6606$. One account is to be chosen at random. What is the probability that the selected account number is $1940$? answer: [ANS]", "answer_v3": [ "0.000151377535573721" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0041", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "Conditional", "Probability" ], "problem_v1": "Scoring a hole-in-one is the greatest shot a golfer can make. Once $6$ professional golfers each made holes-in-one on the $7^{th}$ hole at the same golf course at the same tournament. It has been found that the estimated probability of making a hole-in-one is $\\frac{1}{2753}$ for male professionals. Suppose that a sample of $6$ professional male golfers is randomly selected.\n(a) What is the probability that all of these golfers make a hole-in-one on the $14^{th}$ hole at the same tournament? answer: [ANS]\n(b) What is the probability that none of these golfers make a hole-in-one on the $14^{th}$ hole at the same tournament? answer: [ANS]", "answer_v1": [ "2.29700918414184E-21", "0.997822537582048" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Scoring a hole-in-one is the greatest shot a golfer can make. Once $8$ professional golfers each made holes-in-one on the $4^{th}$ hole at the same golf course at the same tournament. It has been found that the estimated probability of making a hole-in-one is $\\frac{1}{2083}$ for male professionals. Suppose that a sample of $8$ professional male golfers is randomly selected.\n(a) What is the probability that at least one of these golfers makes a hole-in-one on the $11^{th}$ hole at the same tournament? answer: [ANS]\n(b) What is the probability that all of these golfers make a hole-in-one on the $11^{th}$ hole at the same tournament? answer: [ANS]", "answer_v2": [ "0.0038341674258493", "2.82153758918868E-27" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Scoring a hole-in-one is the greatest shot a golfer can make. Once $6$ professional golfers each made holes-in-one on the $5^{th}$ hole at the same golf course at the same tournament. It has been found that the estimated probability of making a hole-in-one is $\\frac{1}{2313}$ for male professionals. Suppose that a sample of $6$ professional male golfers is randomly selected.\n(a) What is the probability that all of these golfers make a hole-in-one on the $13^{th}$ hole at the same tournament? answer: [ANS]\n(b) What is the probability that none of these golfers make a hole-in-one on the $13^{th}$ hole at the same tournament? answer: [ANS]", "answer_v3": [ "6.53049653524992E-21", "0.99740876841642" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0042", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "algebra", "probability" ], "problem_v1": "A poker hand, consisting of 8 cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains 8 hearts. Your answer is: [ANS]", "answer_v1": [ "1.71021229953591E-06" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A poker hand, consisting of 2 cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains 2 hearts. Your answer is: [ANS]", "answer_v2": [ "0.0588235294117647" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A poker hand, consisting of 4 cards, is dealt from a standard deck of 52 cards. Find the probability that the hand contains 4 hearts. Your answer is: [ANS]", "answer_v3": [ "0.00264105642256903" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0043", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "algebra", "probability" ], "problem_v1": "In the 6/39 lottery game, a player selects 6 numbers from 1 to 39. What is the probability of picking the 6 winning numbers? Your answer is: [ANS]", "answer_v1": [ "3.06501854489471E-07" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In the 3/48 lottery game, a player selects 3 numbers from 1 to 48. What is the probability of picking the 3 winning numbers? Your answer is: [ANS]", "answer_v2": [ "5.78168362627197E-05" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In the 4/40 lottery game, a player selects 4 numbers from 1 to 40. What is the probability of picking the 4 winning numbers? Your answer is: [ANS]", "answer_v3": [ "1.09421162052741E-05" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0044", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "algebra", "combination" ], "problem_v1": "6 cards are drawn at random from a standard deck. Find the probability that all the cards are hearts. [ANS]\nFind the probability that all the cards are face cards. [ANS]\nNote: Face cards are kings, queens, and jacks.\nFind the probability that all the cards are even. [ANS]\n(Consider aces to be 1, jacks to be 11, queens to be 12, and kings to be 13)", "answer_v1": [ "8.42890347628413E-05", "4.53864033338376E-05", "0.00661128608562901" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "3 cards are drawn at random from a standard deck. Find the probability that all the cards are hearts. [ANS]\nFind the probability that all the cards are face cards. [ANS]\nNote: Face cards are kings, queens, and jacks.\nFind the probability that all the cards are even. [ANS]\n(Consider aces to be 1, jacks to be 11, queens to be 12, and kings to be 13)", "answer_v2": [ "0.0129411764705882", "0.00995475113122172", "0.0915837104072398" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "4 cards are drawn at random from a standard deck. Find the probability that all the cards are hearts. [ANS]\nFind the probability that all the cards are face cards. [ANS]\nNote: Face cards are kings, queens, and jacks.\nFind the probability that all the cards are even. [ANS]\n(Consider aces to be 1, jacks to be 11, queens to be 12, and kings to be 13)", "answer_v3": [ "0.00264105642256903", "0.00182842367716317", "0.0392501616031028" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0045", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "algebra", "combination" ], "problem_v1": "An algebra class has 20 students and 20 desks. For the sake of variety, students change the seating arrangement each day. How many days must pass before the class must repeat a seating arrangement? [ANS] days must pass before a seating arrangement is repeated.\nSuppose the desks are arranged in rows of 4. How many seating arrangements are there that put Larry, Moe, Curly, and Shemp in the front seats?\nThere are [ANS] seating arrangements that put them in the front seats.\nWhat is the probability that Larry, Moe, Curly and Shemp are sitting in the front seats?\nThe probability is [ANS].", "answer_v1": [ "2432902008176640000", "502146957312000", "0.000206398348813209" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An algebra class has 8 students and 8 desks. For the sake of variety, students change the seating arrangement each day. How many days must pass before the class must repeat a seating arrangement? [ANS] days must pass before a seating arrangement is repeated.\nSuppose the desks are arranged in rows of 4. How many seating arrangements are there that put Larry, Moe, Curly, and Shemp in the front seats?\nThere are [ANS] seating arrangements that put them in the front seats.\nWhat is the probability that Larry, Moe, Curly and Shemp are sitting in the front seats?\nThe probability is [ANS].", "answer_v2": [ "40320", "576", "0.0142857142857143" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An algebra class has 12 students and 12 desks. For the sake of variety, students change the seating arrangement each day. How many days must pass before the class must repeat a seating arrangement? [ANS] days must pass before a seating arrangement is repeated.\nSuppose the desks are arranged in rows of 4. How many seating arrangements are there that put Larry, Moe, Curly, and Shemp in the front seats?\nThere are [ANS] seating arrangements that put them in the front seats.\nWhat is the probability that Larry, Moe, Curly and Shemp are sitting in the front seats?\nThe probability is [ANS].", "answer_v3": [ "479001600", "967680", "0.00202020202020202" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0046", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "logarithms", "exponentials", "probability" ], "problem_v1": "Suppose a number is chosen at random from the set {0,1,2,3,...,688}. What is the probability that the number is a perfect cube? The probability of choosing a perfect cube is [ANS]\nNote: Your answer must be a fraction or a decimal number.", "answer_v1": [ "0.0130624092888244" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose a number is chosen at random from the set {0,1,2,3,...,111}. What is the probability that the number is a perfect cube? The probability of choosing a perfect cube is [ANS]\nNote: Your answer must be a fraction or a decimal number.", "answer_v2": [ "0.0446428571428571" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose a number is chosen at random from the set {0,1,2,3,...,309}. What is the probability that the number is a perfect cube? The probability of choosing a perfect cube is [ANS]\nNote: Your answer must be a fraction or a decimal number.", "answer_v3": [ "0.0225806451612903" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0047", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "algebra", "combination" ], "problem_v1": "A bag contains 9 red marbles, 8 white marbles, and 8 blue marbles. You draw 5 marbles out at random, without replacement. What is the probability that all the marbles are red?\nThe probability that all the marbles are red is [ANS].\nWhat is the probability that exactly two of the marbles are red? The probability that exactly two of the marbles are red is [ANS].\nWhat is the probability that none of the marbles are red? The probability of picking no red marbles is [ANS].", "answer_v1": [ "126/53130", "20160/53130", "4368/53130" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A bag contains 7 red marbles, 10 white marbles, and 5 blue marbles. You draw 3 marbles out at random, without replacement. What is the probability that all the marbles are red?\nThe probability that all the marbles are red is [ANS].\nWhat is the probability that exactly two of the marbles are red? The probability that exactly two of the marbles are red is [ANS].\nWhat is the probability that none of the marbles are red? The probability of picking no red marbles is [ANS].", "answer_v2": [ "35/1540", "315/1540", "455/1540" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A bag contains 8 red marbles, 8 white marbles, and 6 blue marbles. You draw 3 marbles out at random, without replacement. What is the probability that all the marbles are red?\nThe probability that all the marbles are red is [ANS].\nWhat is the probability that exactly two of the marbles are red? The probability that exactly two of the marbles are red is [ANS].\nWhat is the probability that none of the marbles are red? The probability of picking no red marbles is [ANS].", "answer_v3": [ "56/1540", "392/1540", "364/1540" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0048", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "logarithms", "exponentials", "algebra", "probability" ], "problem_v1": "What is the probability that if 12 letters are typed, no letters are repeated? The probability that no letters are repeated is [ANS].", "answer_v1": [ "0.0484764154838288" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the probability that if 4 letters are typed, no letters are repeated? The probability that no letters are repeated is [ANS].", "answer_v2": [ "0.785161583978152" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the probability that if 7 letters are typed, no letters are repeated? The probability that no letters are repeated is [ANS].", "answer_v3": [ "0.41277270345688" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0049", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "statistics", "probability" ], "problem_v1": "Government data assign a single cause for each death that occurs in the United States. In a certain city, the data show that the probability is 0.38 that a randomly chosen death was due to cardiovascular (mainly heart) disease, and 0.25 that it was due to cancer.\n(a) The probability that a death was due either to cardiovascular disease or to cancer is [ANS]. (b) The probability that the death was due to some other cause is [ANS].", "answer_v1": [ "0.38+0.25", "0.37" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Government data assign a single cause for each death that occurs in the United States. In a certain city, the data show that the probability is 0.3 that a randomly chosen death was due to cardiovascular (mainly heart) disease, and 0.25 that it was due to cancer.\n(a) The probability that a death was due either to cardiovascular disease or to cancer is [ANS]. (b) The probability that the death was due to some other cause is [ANS].", "answer_v2": [ "0.3+0.25", "0.45" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Government data assign a single cause for each death that occurs in the United States. In a certain city, the data show that the probability is 0.33 that a randomly chosen death was due to cardiovascular (mainly heart) disease, and 0.25 that it was due to cancer.\n(a) The probability that a death was due either to cardiovascular disease or to cancer is [ANS]. (b) The probability that the death was due to some other cause is [ANS].", "answer_v3": [ "0.33+0.25", "0.42" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0050", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "statistics", "probability" ], "problem_v1": "A basketball player makes 180 out of 220 free throws. We would estimate the probability that the player makes the next free throw to be [ANS].", "answer_v1": [ "180/220" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A basketball player makes 60 out of 110 free throws. We would estimate the probability that the player makes the next free throw to be [ANS].", "answer_v2": [ "60/110" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A basketball player makes 110 out of 150 free throws. We would estimate the probability that the player makes the next free throw to be [ANS].", "answer_v3": [ "110/150" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0051", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "statistics", "probability" ], "problem_v1": "$\\begin{array}{cccccc}\\hline x & Attend a 4-year college & Attend a junior college & Attend a technical school & Train as an apprentice & No formal training after high school \\\\ \\hline P(x) & 0.2 & 0.2 & 0.1 & 0.1 & 0.4 \\\\ \\hline \\end{array}$ At a certain high school, if a student is selected at random and asked what they plan to do after graduating, the probability distribution for their response is given above, determine the following:\n(a) P(Do not attend a 4-year college) $=$ [ANS]\n(b) P(Attend a college) $=$ [ANS]\n(c) P(Receive some sort of training after high school) $=$ [ANS]", "answer_v1": [ "0.8", "0.4", "0.6" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "$\\begin{array}{cccccc}\\hline x & Attend a 4-year college & Attend a junior college & Attend a technical school & Train as an apprentice & No formal training after high school \\\\ \\hline P(x) & 0.1 & 0.2 & 0.1 & 0.1 & 0.5 \\\\ \\hline \\end{array}$ At a certain high school, if a student is selected at random and asked what they plan to do after graduating, the probability distribution for their response is given above, determine the following:\n(a) P(Do not attend a 4-year college) $=$ [ANS]\n(b) P(Receive training after high school but not at a college) $=$ [ANS]\n(c) P(Do not attend any college) $=$ [ANS]", "answer_v2": [ "0.9", "0.2", "0.7" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "$\\begin{array}{cccccc}\\hline x & Attend a 4-year college & Attend a junior college & Attend a technical school & Train as an apprentice & No formal training after high school \\\\ \\hline P(x) & 0.1 & 0.2 & 0.1 & 0.1 & 0.5 \\\\ \\hline \\end{array}$ At a certain high school, if a student is selected at random and asked what they plan to do after graduating, the probability distribution for their response is given above, determine the following:\n(a) P(Receive some sort of training after high school) $=$ [ANS]\n(b) P(Do not attend any college) $=$ [ANS]\n(c) P(Do not attend a 4-year college) $=$ [ANS]", "answer_v3": [ "0.5", "0.7", "0.9" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0052", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "statistics", "probability" ], "problem_v1": "Choose a student in grades 9 to 12 at random and ask if he or she is studying a language other than English. Here is the distribution of the students:\n$\\begin{array}{cccccc}\\hline Language & Spanish & French & German & All Others & None \\\\ \\hline Probability & 0.27 & 0.08 & 0.03 & 0.03 & 0.59 \\\\ \\hline \\end{array}$\n(a) What is the probability that a randomly chosen student is, in fact, studying a language other than English? ANSWER [ANS]\n(b) What is the probability that a randomly chosen student is studying French, German, or Spanish? ANSWER [ANS]\n(c) What is the probability that a randomly chosen student is studying a language besides English, but not German? ANSWER [ANS]", "answer_v1": [ "0.41", "0.38", "0.38" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Choose a student in grades 9 to 12 at random and ask if he or she is studying a language other than English. Here is the distribution of the students:\n$\\begin{array}{cccccc}\\hline Language & Spanish & French & German & All Others & None \\\\ \\hline Probability & 0.2 & 0.1 & 0.01 & 0.02 & 0.67 \\\\ \\hline \\end{array}$\n(a) What is the probability that a randomly chosen student is, in fact, studying a language other than English? ANSWER [ANS]\n(b) What is the probability that a randomly chosen student is studying French, German, or Spanish? ANSWER [ANS]\n(c) What is the probability that a randomly chosen student is studying a language besides English, but not German? ANSWER [ANS]", "answer_v2": [ "0.33", "0.31", "0.32" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Choose a student in grades 9 to 12 at random and ask if he or she is studying a language other than English. Here is the distribution of the students:\n$\\begin{array}{cccccc}\\hline Language & Spanish & French & German & All Others & None \\\\ \\hline Probability & 0.23 & 0.08 & 0.02 & 0.03 & 0.64 \\\\ \\hline \\end{array}$\n(a) What is the probability that a randomly chosen student is, in fact, studying a language other than English? ANSWER [ANS]\n(b) What is the probability that a randomly chosen student is studying French, German, or Spanish? ANSWER [ANS]\n(c) What is the probability that a randomly chosen student is studying a language besides English, but not German? ANSWER [ANS]", "answer_v3": [ "0.36", "0.33", "0.34" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0053", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "statistics", "probability" ], "problem_v1": "Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. [ANS] 1. This event is certain. It will occur on every trial. [ANS] 2. This event is impossible. It will never happen. [ANS] 3. This event will occur more often than not. [ANS] 4. This event is very likely to occur.\nA. 1 B. 0.96 C. 0 D. 0.58", "answer_v1": [ "A", "C", "D", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. [ANS] 1. This event is very likely to occur. [ANS] 2. This event is very unlikely to happen, but it will occur once in a while in a long sequence of trials. [ANS] 3. This event will occur a little less than half the time over a long sequence of trials. [ANS] 4. This event will occur more often than not.\nA. 0.55 B. 0.93 C. 0.44 D. 0.01", "answer_v2": [ "B", "D", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. [ANS] 1. This event is certain. It will occur on every trial. [ANS] 2. This event is very likely to occur. [ANS] 3. This event will occur a little less than half the time over a long sequence of trials. [ANS] 4. This event will occur more often than not.\nA. 0.94 B. 1 C. 0.56 D. 0.43", "answer_v3": [ "B", "A", "D", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0054", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "random sampling", "probability distributions" ], "problem_v1": "The age distribution for senators in the 104th U.S. Congress was as follows:\n$\\begin{array}{cccccc}\\hline age & under 40 & 40-49 & 50-59 & 60-69 & 70 and over \\\\ \\hline no. of senators & 1 & 14 & 41 & 27 & 17 \\\\ \\hline \\end{array}$\nConsider the following four events:\n$A$=event the senator is under 40 $B$=event the senator is in his or her 50s $C$=event the senator is 40 or older $D$=event the senator is under 60\nFind the probability of each event given below:\n(a) $A$ and $D$=[ANS]\n(b) not $A$=[ANS]\n(c) $C$ and $D$=[ANS]\n(d) not $C$=[ANS]\n(e) $C$=[ANS]", "answer_v1": [ "0.01", "0.99", "0.55", "0.01", "0.99" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The age distribution for senators in the 104th U.S. Congress was as follows:\n$\\begin{array}{cccccc}\\hline age & under 40 & 40-49 & 50-59 & 60-69 & 70 and over \\\\ \\hline no. of senators & 1 & 14 & 41 & 27 & 17 \\\\ \\hline \\end{array}$\nConsider the following four events:\n$A$=event the senator is under 40 $B$=event the senator is in his or her 50s $C$=event the senator is 40 or older $D$=event the senator is under 60\nFind the probability of each event given below:\n(a) $B$=[ANS]\n(b) $A$ or $C$=[ANS]\n(c) $C$=[ANS]\n(d) not $D$=[ANS]\n(e) $A$ or $D$=[ANS]", "answer_v2": [ "0.41", "1", "0.99", "0.44", "0.56" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The age distribution for senators in the 104th U.S. Congress was as follows:\n$\\begin{array}{cccccc}\\hline age & under 40 & 40-49 & 50-59 & 60-69 & 70 and over \\\\ \\hline no. of senators & 1 & 14 & 41 & 27 & 17 \\\\ \\hline \\end{array}$\nConsider the following four events:\n$A$=event the senator is under 40 $B$=event the senator is in his or her 50s $C$=event the senator is 40 or older $D$=event the senator is under 60\nFind the probability of each event given below:\n(a) not $B$=[ANS]\n(b) $B$ and $D$=[ANS]\n(c) $D$=[ANS]\n(d) $C$ and $D$=[ANS]\n(e) $B$=[ANS]", "answer_v3": [ "0.59", "0.41", "0.56", "0.55", "0.41" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0055", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "random sampling", "probability distributions" ], "problem_v1": "$\\begin{array}{ccccccccc}\\hline rooms & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8+\\\\ \\hline no. of units (in thousands) & 47 & 140 & 1170 & 2350 & 2450 & 2130 & 1370 & 1560 \\\\ \\hline \\end{array}$\nThe table above provides information on housing units in some part of the U.S.\nConsider the following five events:\nA=the unit has at most four rooms B=the unit has at least two rooms C=the unit has between five and seven rooms inclusive D=the unit has more than seven rooms E=the unit has less than three rooms\nDetermine the number of outcomes (in thousands) that comprise each event given below.\n(a) A and D: [ANS]\n(b) B and E: [ANS]\n(c) A or D: [ANS]\n(d) not C: [ANS]\n(e) not D: [ANS]", "answer_v1": [ "0", "140", "5267", "5267", "9657" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "$\\begin{array}{ccccccccc}\\hline rooms & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8+\\\\ \\hline no. of units (in thousands) & 47 & 140 & 1170 & 2350 & 2450 & 2130 & 1370 & 1560 \\\\ \\hline \\end{array}$\nThe table above provides information on housing units in some part of the U.S.\nConsider the following five events:\nA=the unit has at most four rooms B=the unit has at least two rooms C=the unit has between five and seven rooms inclusive D=the unit has more than seven rooms E=the unit has less than three rooms\nDetermine the number of outcomes (in thousands) that comprise each event given below.\n(a) not A: [ANS]\n(b) A or C: [ANS]\n(c) not D: [ANS]\n(d) C and D: [ANS]\n(e) B and C: [ANS]", "answer_v2": [ "7510", "9657", "9657", "0", "5950" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "$\\begin{array}{ccccccccc}\\hline rooms & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8+\\\\ \\hline no. of units (in thousands) & 47 & 140 & 1170 & 2350 & 2450 & 2130 & 1370 & 1560 \\\\ \\hline \\end{array}$\nThe table above provides information on housing units in some part of the U.S.\nConsider the following five events:\nA=the unit has at most four rooms B=the unit has at least two rooms C=the unit has between five and seven rooms inclusive D=the unit has more than seven rooms E=the unit has less than three rooms\nDetermine the number of outcomes (in thousands) that comprise each event given below.\n(a) not E: [ANS]\n(b) not C: [ANS]\n(c) not D: [ANS]\n(d) A or D: [ANS]\n(e) not B: [ANS]", "answer_v3": [ "11030", "5267", "9657", "5267", "47" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0056", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "probability" ], "problem_v1": "Consider a sample space of three outcomes A,B,and C. Which of the following represent legitimate probability models? [ANS] A. P(A)=0.4, P(B)=0.5, P(C)=0.1 B. P(A)=0.1, P(B)=0.4, P(C)=0.5 C. P(A)=0.9, P(B)=0.1, P(C)=0.2 D. P(A)=0.4, P(B)=0.1,P(C)=0.4 E. P(A)=-0.8, P(B)=0.8, P(C)=1", "answer_v1": [ "AB" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Consider a sample space of three outcomes A,B,and C. Which of the following represent legitimate probability models? [ANS] A. P(A)=0.2, P(B)=0.6,P(C)=0.1 B. P(A)=0.1, P(B)=0.1, P(C)=1 C. P(A)=-0.6, P(B)=0.8, P(C)=0.8 D. P(A)=0.2, P(B)=0.4, P(C)=0.4 E. P(A)=0.2, P(B)=0.2, P(C)=0.6", "answer_v2": [ "DE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Consider a sample space of three outcomes A,B,and C. Which of the following represent legitimate probability models? [ANS] A. P(A)=0.4, P(B)=0,P(C)=0.5 B. P(A)=0.2, P(B)=0.5, P(C)=0.5 C. P(A)=0.6, P(B)=0.1, P(C)=0.3 D. P(A)=-0.9, P(B)=1, P(C)=0.9 E. P(A)=0.1, P(B)=0.5, P(C)=0.4", "answer_v3": [ "CE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0057", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "probability" ], "problem_v1": "A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Flush (5 nonconsecutive cards each of the same suit). Answer: [ANS]", "answer_v1": [ "0.00196540154523348" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Royal flush (10, J, Q, K, A all of the same suit). Answer: [ANS]", "answer_v2": [ "1.53907716932927E-06" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Straight flush (5cards in a sequence in a single suit, but not a royal flush). Answer: [ANS]", "answer_v3": [ "1.38516945239634E-05" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0059", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "union", "intersection", "complement" ], "problem_v1": "The data below was obtained from a random survey of $471$ people. The participants were asked their political party and whether they approve, disapprove, or have no opinion of how the president is doing his job. The results of the survey are as follows:\n$\\begin{array}{ccccc}\\hline & Democrat & Republican & Independent & Total \\\\ \\hline Approve & 107 & 94 & 8 & 209 \\\\ \\hline Disapprove & 118 & 84 & 8 & 210 \\\\ \\hline Undecided & 26 & 26 & 3 & 52 \\\\ \\hline Total & 251 & 204 & 19 & 471 \\\\ \\hline \\end{array}$\nIf a person if selected at random, what is the empirical probability that the person approves of the job the president is doing or is an independent? [ANS]", "answer_v1": [ "0.467091" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The data below was obtained from a random survey of $457$ people. The participants were asked their political party and whether they approve, disapprove, or have no opinion of how the president is doing his job. The results of the survey are as follows:\n$\\begin{array}{ccccc}\\hline & Democrat & Republican & Independent & Total \\\\ \\hline Approve & 56 & 120 & 5 & 181 \\\\ \\hline Disapprove & 87 & 136 & 8 & 231 \\\\ \\hline Undecided & 22 & 23 & 4 & 45 \\\\ \\hline Total & 165 & 279 & 17 & 457 \\\\ \\hline \\end{array}$\nIf a person if selected at random, what is the empirical probability that the person approves of the job the president is doing or is an independent? [ANS]", "answer_v2": [ "0.422319" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The data below was obtained from a random survey of $421$ people. The participants were asked their political party and whether they approve, disapprove, or have no opinion of how the president is doing his job. The results of the survey are as follows:\n$\\begin{array}{ccccc}\\hline & Democrat & Republican & Independent & Total \\\\ \\hline Approve & 73 & 96 & 6 & 175 \\\\ \\hline Disapprove & 104 & 76 & 8 & 188 \\\\ \\hline Undecided & 28 & 30 & 5 & 58 \\\\ \\hline Total & 205 & 202 & 19 & 421 \\\\ \\hline \\end{array}$\nIf a person if selected at random, what is the empirical probability that the person approves of the job the president is doing or is an independent? [ANS]", "answer_v3": [ "0.446556" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0060", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "probability", "events" ], "problem_v1": "$7$ cards are drawn simultaneously from a standard deck of $52$ cards. a) What is the size of the sample space? [ANS]\nb) What is the probability that at least $6$ of cards drawn are red? [ANS]", "answer_v1": [ "1.33785E+08", "0.0496603" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "$7$ cards are drawn simultaneously from a standard deck of $52$ cards. a) What is the size of the sample space? [ANS]\nb) What is the probability that at least $3$ of cards drawn are red? [ANS]", "answer_v2": [ "1.33785E+08", "0.790542" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "$7$ cards are drawn simultaneously from a standard deck of $52$ cards. a) What is the size of the sample space? [ANS]\nb) What is the probability that at least $4$ of cards drawn are red? [ANS]", "answer_v3": [ "1.33785E+08", "0.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0061", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "What is the probability that a 9-digit phone number contains at least one 6? (Repetition of numbers and lead zero are allowed). Answer: [ANS]", "answer_v1": [ "0.612579511" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the probability that a 6-digit phone number contains at least one 9? (Repetition of numbers and lead zero are allowed). Answer: [ANS]", "answer_v2": [ "0.468559" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the probability that a 7-digit phone number contains at least one 6? (Repetition of numbers and lead zero are allowed). Answer: [ANS]", "answer_v3": [ "0.5217031" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0062", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "algebra", "probability" ], "problem_v1": "A committee of four is chosen at random from a group of $8$ women and $4$ men. Find the probability that the committee contains at least one man. Answer: [ANS]", "answer_v1": [ "1-8/12*(8-1)/(12-1)*(8-2)/(12-2)*(8-3)/(12-3)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A committee of four is chosen at random from a group of $6$ women and $5$ men. Find the probability that the committee contains at least one man. Answer: [ANS]", "answer_v2": [ "1-6/11*(6-1)/(11-1)*(6-2)/(11-2)*(6-3)/(11-3)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A committee of four is chosen at random from a group of $6$ women and $4$ men. Find the probability that the committee contains at least one man. Answer: [ANS]", "answer_v3": [ "1-6/10*(6-1)/(10-1)*(6-2)/(10-2)*(6-3)/(10-3)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0063", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "algebra", "probability", "binomial theorem" ], "problem_v1": "It is known that a certain drug causes side effects in $12$ \\% of patients. If we consider a random sampling of $15$ patients, what is the probability less than two have the side effect? Answer: [ANS]", "answer_v1": [ "0.88^15+15*0.88^14*0.12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "It is known that a certain drug causes side effects in $8$ \\% of patients. If we consider a random sampling of $18$ patients, what is the probability less than two have the side effect? Answer: [ANS]", "answer_v2": [ "0.92^18+18*0.92^17*0.08" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "It is known that a certain drug causes side effects in $8$ \\% of patients. If we consider a random sampling of $15$ patients, what is the probability less than two have the side effect? Answer: [ANS]", "answer_v3": [ "0.92^15+15*0.92^14*0.08" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0064", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "algebra", "probability" ], "problem_v1": "Suppose that $29$ slips of paper numbered $1$ to $29$, inclusive, are put into a hat. Then one is drawn out at random. Find the probability of each of the following events: 1. The slip with a 12 is drawn. Answer: [ANS]\n2. A slip with an even number on it is drawn. Answer: [ANS]\n3. A slip with a prime number on it is drawn. Answer: [ANS]\n4. A slip with a multiple of five on it is drawn. Answer: [ANS]", "answer_v1": [ "1/29", "(29-1)/2/29", "10/29", "5/29" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose that $21$ slips of paper numbered $1$ to $21$, inclusive, are put into a hat. Then one is drawn out at random. Find the probability of each of the following events: 1. The slip with a 12 is drawn. Answer: [ANS]\n2. A slip with an even number on it is drawn. Answer: [ANS]\n3. A slip with a prime number on it is drawn. Answer: [ANS]\n4. A slip with a multiple of five on it is drawn. Answer: [ANS]", "answer_v2": [ "1/21", "(21-1)/2/21", "8/21", "4/21" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose that $23$ slips of paper numbered $1$ to $23$, inclusive, are put into a hat. Then one is drawn out at random. Find the probability of each of the following events: 1. The slip with a 12 is drawn. Answer: [ANS]\n2. A slip with an even number on it is drawn. Answer: [ANS]\n3. A slip with a prime number on it is drawn. Answer: [ANS]\n4. A slip with a multiple of five on it is drawn. Answer: [ANS]", "answer_v3": [ "1/23", "(23-1)/2/23", "9/23", "4/23" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0065", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "1", "keywords": [ "algebra", "probability" ], "problem_v1": "The probability that a certain horse will win the Kentucky Derby is $ \\frac{1}{25}.$ What is the probability that it will lose the race? Answer: [ANS]", "answer_v1": [ "1-1/25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The probability that a certain horse will win the Kentucky Derby is $ \\frac{1}{5}.$ What is the probability that it will lose the race? Answer: [ANS]", "answer_v2": [ "1-1/5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The probability that a certain horse will win the Kentucky Derby is $ \\frac{1}{10}.$ What is the probability that it will lose the race? Answer: [ANS]", "answer_v3": [ "1-1/10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0066", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "algebra", "probability", "binomial theorem" ], "problem_v1": "Trees planted by a landscaping firm have a $95$ \\% one-year survival rate, If they plant $15$ trees in a park, what is the following probabilities: 1. All the trees survive one year. Answer: [ANS]\n2. At least $13$ trees survive one year. Answer: [ANS]", "answer_v1": [ "0.95^15", "0.95^15+15*0.95^14*(1-0.95)+15*(15-1)/2*0.95^13*(1-0.95)^2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Trees planted by a landscaping firm have a $90$ \\% one-year survival rate, If they plant $16$ trees in a park, what is the following probabilities: 1. All the trees survive one year. Answer: [ANS]\n2. At least $14$ trees survive one year. Answer: [ANS]", "answer_v2": [ "0.9^16", "0.9^16+16*0.9^15*(1-0.9)+16*(16-1)/2*0.9^14*(1-0.9)^2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Trees planted by a landscaping firm have a $90$ \\% one-year survival rate, If they plant $15$ trees in a park, what is the following probabilities: 1. All the trees survive one year. Answer: [ANS]\n2. At least $13$ trees survive one year. Answer: [ANS]", "answer_v3": [ "0.9^15", "0.9^15+15*0.9^14*(1-0.9)+15*(15-1)/2*0.9^13*(1-0.9)^2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0067", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "algebra", "probability" ], "problem_v1": "April, Bill, Candace, and Bobby are to be seated at random in a row of $7$ chairs. What is the probability that April and Bobby will occupy the seats at the end of the row? Answer: [ANS]", "answer_v1": [ "2/7*1/(7-1)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "April, Bill, Candace, and Bobby are to be seated at random in a row of $4$ chairs. What is the probability that April and Bobby will occupy the seats at the end of the row? Answer: [ANS]", "answer_v2": [ "2/4*1/(4-1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "April, Bill, Candace, and Bobby are to be seated at random in a row of $5$ chairs. What is the probability that April and Bobby will occupy the seats at the end of the row? Answer: [ANS]", "answer_v3": [ "2/5*1/(5-1)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0068", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "3", "keywords": [ "Probability", "elementary probability", "equally likely events", "sampling without replacement", "lottery", "finding probabilities of winning in a N choose 5 lottery" ], "problem_v1": "The country of Statland runs a national lottery for its people. In this lottery, an urn contains 42 balls that are individually numbered 1, 2,..., 42. Each week, 5 balls are drawn from the urn without replacement. Gamblers pay a small sum to predict these five numbers and the jackpot is won by anyone who has chosen all five correctly. A runners-up prize is won by anyone who correctly selected four of the five numbers. You have decided to play the Statland lottery and have chosen your five numbers at random from the 42 available. Give your answer to seven decimals in each part. (You may also express your answer exactly as a fraction; for example, if the exact answer were 1/9 then both \"1/9\" and \"0.1111111\" would be correct).\nPart a) What is the probability that you win the jackpot? [ANS]\nPart b) What is the probability that you win a runners-up prize? [ANS]", "answer_v1": [ "1.2E-06", "0.0002175" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The country of Statland runs a national lottery for its people. In this lottery, an urn contains 36 balls that are individually numbered 1, 2,..., 36. Each week, 5 balls are drawn from the urn without replacement. Gamblers pay a small sum to predict these five numbers and the jackpot is won by anyone who has chosen all five correctly. A runners-up prize is won by anyone who correctly selected four of the five numbers. You have decided to play the Statland lottery and have chosen your five numbers at random from the 36 available. Give your answer to seven decimals in each part. (You may also express your answer exactly as a fraction; for example, if the exact answer were 1/9 then both \"1/9\" and \"0.1111111\" would be correct).\nPart a) What is the probability that you win the jackpot? [ANS]\nPart b) What is the probability that you win a runners-up prize? [ANS]", "answer_v2": [ "2.7E-06", "0.0004111" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The country of Statland runs a national lottery for its people. In this lottery, an urn contains 38 balls that are individually numbered 1, 2,..., 38. Each week, 5 balls are drawn from the urn without replacement. Gamblers pay a small sum to predict these five numbers and the jackpot is won by anyone who has chosen all five correctly. A runners-up prize is won by anyone who correctly selected four of the five numbers. You have decided to play the Statland lottery and have chosen your five numbers at random from the 38 available. Give your answer to seven decimals in each part. (You may also express your answer exactly as a fraction; for example, if the exact answer were 1/9 then both \"1/9\" and \"0.1111111\" would be correct).\nPart a) What is the probability that you win the jackpot? [ANS]\nPart b) What is the probability that you win a runners-up prize? [ANS]", "answer_v3": [ "2E-06", "0.0003287" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0069", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [], "problem_v1": "Two cards are drawn at random from a pack without replacement. What is the probability that the first is an Ace but the second is not an Ace? [ANS] A. 16/221 B. 48/51 C. 49/64 D. 2/507 E. 12/13", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Two cards are drawn at random from a pack without replacement. What is the probability that the first is an Ace and the second is a Queen? [ANS] A. 2/52 B. 5/64 C. 1/169 D. 4/663 E. 2/13", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Two cards are drawn at random from a pack without replacement. What is the probability that the first is an Ace and the second is a Queen? [ANS] A. 2/13 B. 2/52 C. 1/169 D. 5/64 E. 4/663", "answer_v3": [ "E" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0070", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "", "keywords": [], "problem_v1": "In a large city, 37\\% of all restaurants accept both master and visa credit cards, and 50\\% accept master cards and 60\\% accept visa cards. A tourist visiting the city picks at random a restaurant at which to have lunch. Define the following events: $M$={the randomly chosen restaurant accepts master credit cards}, $V$={the randomly chosen restaurant accepts visa credit cards}.\nPart a Are $M$ and $V$ disjoint? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart b Are $M$ and $V$ independent? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart c Is $M$ complement of $V$? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart d Which of the following shows independence between two events $M$ and $V$? [CHECK ALL THAT APPLY] [ANS] A. $P(V \\text{given} M)$ is equal to $P(V)$. B. The events $M$ and $V$ are disjoint. C. $P(M \\text{given} V)$ is equal to $P(M)$. D. The events $M$ and $V$ are not disjoint. E. $P(M \\text{given} V)$ is equal to $P(V)$. F. $P(M \\text{and} V)$ is equal to $P(M) \\times P(V)$.\nPart e What is the probability that a given restaurant does not accept master or visa credit cards? [ANS]", "answer_v1": [ "A", "A", "A", "ACF", "0.27" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCM", "NV" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E", "F" ], [] ], "problem_v2": "In a large city, 37\\% of all restaurants accept both master and visa credit cards, and 50\\% accept master cards and 60\\% accept visa cards. A tourist visiting the city picks at random a restaurant at which to have lunch. Define the following events: $M$={the randomly chosen restaurant accepts master credit cards}, $V$={the randomly chosen restaurant accepts visa credit cards}.\nPart a Are $M$ and $V$ disjoint? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart b Are $M$ and $V$ independent? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart c Is $M$ complement of $V$? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart d Which of the following shows independence between two events $M$ and $V$? [CHECK ALL THAT APPLY] [ANS] A. $P(M \\text{given} V)$ is equal to $P(V)$. B. $P(V \\text{given} M)$ is equal to $P(V)$. C. $P(M \\text{and} V)$ is equal to $P(M) \\times P(V)$. D. The events $M$ and $V$ are not disjoint. E. $P(M \\text{given} V)$ is equal to $P(M)$. F. The events $M$ and $V$ are disjoint.\nPart e What is the probability that a given restaurant does not accept master or visa credit cards? [ANS]", "answer_v2": [ "A", "A", "A", "BCE", "0.27" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCM", "NV" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E", "F" ], [] ], "problem_v3": "In a large city, 37\\% of all restaurants accept both master and visa credit cards, and 50\\% accept master cards and 60\\% accept visa cards. A tourist visiting the city picks at random a restaurant at which to have lunch. Define the following events: $M$={the randomly chosen restaurant accepts master credit cards}, $V$={the randomly chosen restaurant accepts visa credit cards}.\nPart a Are $M$ and $V$ disjoint? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart b Are $M$ and $V$ independent? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart c Is $M$ complement of $V$? [ANS] A. No. B. Yes. C. Insufficient information to tell.\nPart d Which of the following shows independence between two events $M$ and $V$? [CHECK ALL THAT APPLY] [ANS] A. $P(M \\text{given} V)$ is equal to $P(V)$. B. $P(M \\text{and} V)$ is equal to $P(M) \\times P(V)$. C. $P(V \\text{given} M)$ is equal to $P(V)$. D. The events $M$ and $V$ are not disjoint. E. The events $M$ and $V$ are disjoint. F. $P(M \\text{given} V)$ is equal to $P(M)$.\nPart e What is the probability that a given restaurant does not accept master or visa credit cards? [ANS]", "answer_v3": [ "A", "A", "A", "BCF", "0.27" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCM", "NV" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C", "D", "E", "F" ], [] ] }, { "id": "Probability_0071", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "algebra", "inequality", "fraction" ], "problem_v1": "Events $A$ and $B$ are independent. $P(A)=0.4$ and $P(B)=0.4$. Find $P(A \\cup B)$ to two decimal places.\n$P(A\\cup B)=$ [ANS]", "answer_v1": [ "0.64" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Events $A$ and $B$ are independent. $P(A)=0.2$ and $P(B)=0.4$. Find $P(A \\cup B)$ to two decimal places.\n$P(A\\cup B)=$ [ANS]", "answer_v2": [ "0.52" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Events $A$ and $B$ are independent. $P(A)=0.3$ and $P(B)=0.6$. Find $P(A \\cup B)$ to two decimal places.\n$P(A\\cup B)=$ [ANS]", "answer_v3": [ "0.72" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0072", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "According to the U.S. National Center for Education Statistics, there are more than 63 million American workers 18 years old and over who use computers at work. From this study, which was conducted in 1994 and 1998 (Source: Statistical Abstract of the United States, 2000, Table 690), the following table of joint probabilities was developed.\n\\begin{array}{c||c|c} & \\mbox{Uses Spreadsheet} & \\mbox{No Spreadsheet Use} \\\\ \\hline \\mbox{Female} & 0.33 & 0.25 \\\\ \\hline \\mbox{Male} & 0.24 & 0.18 \\end{array} A. What proportion of workers use a spreadsheet? [ANS]\nB. What proportion of male workers use a spreadsheet? [ANS]\nC. What proportion of spreadsheet users are female? [ANS]", "answer_v1": [ "0.57", "0.571428571428571", "0.578947368421053" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "According to the U.S. National Center for Education Statistics, there are more than 63 million American workers 18 years old and over who use computers at work. From this study, which was conducted in 1994 and 1998 (Source: Statistical Abstract of the United States, 2000, Table 690), the following table of joint probabilities was developed.\n\\begin{array}{c||c|c} & \\mbox{Uses Spreadsheet} & \\mbox{No Spreadsheet Use} \\\\ \\hline \\mbox{Female} & 0.28 & 0.22 \\\\ \\hline \\mbox{Male} & 0.26 & 0.24 \\end{array} A. What proportion of workers use a spreadsheet? [ANS]\nB. What proportion of male workers use a spreadsheet? [ANS]\nC. What proportion of spreadsheet users are female? [ANS]", "answer_v2": [ "0.54", "0.52", "0.518518518518518" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "According to the U.S. National Center for Education Statistics, there are more than 63 million American workers 18 years old and over who use computers at work. From this study, which was conducted in 1994 and 1998 (Source: Statistical Abstract of the United States, 2000, Table 690), the following table of joint probabilities was developed.\n\\begin{array}{c||c|c} & \\mbox{Uses Spreadsheet} & \\mbox{No Spreadsheet Use} \\\\ \\hline \\mbox{Female} & 0.3 & 0.23 \\\\ \\hline \\mbox{Male} & 0.24 & 0.23 \\end{array} A. What proportion of workers use a spreadsheet? [ANS]\nB. What proportion of male workers use a spreadsheet? [ANS]\nC. What proportion of spreadsheet users are female? [ANS]", "answer_v3": [ "0.54", "0.51063829787234", "0.555555555555556" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0073", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability" ], "problem_v1": "Three contractors (call them A, B, and C) bid on a project to build an addition to the UVA Rotunda. Suppose that you believe that Contractor A is 5 times more likely to win than Contractor B, who in turn is 6 times more likely to win than Contractor C. What are each of their probabilities of winning? P(A Wins)=[ANS]\nP(B Wins)=[ANS]\nP(C Wins)=[ANS]", "answer_v1": [ "0.810810810810811", "0.162162162162162", "0.027027027027027" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Three contractors (call them A, B, and C) bid on a project to build an addition to the UVA Rotunda. Suppose that you believe that Contractor A is 2 times more likely to win than Contractor B, who in turn is 8 times more likely to win than Contractor C. What are each of their probabilities of winning? P(A Wins)=[ANS]\nP(B Wins)=[ANS]\nP(C Wins)=[ANS]", "answer_v2": [ "0.64", "0.32", "0.04" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Three contractors (call them A, B, and C) bid on a project to build an addition to the UVA Rotunda. Suppose that you believe that Contractor A is 3 times more likely to win than Contractor B, who in turn is 6 times more likely to win than Contractor C. What are each of their probabilities of winning? P(A Wins)=[ANS]\nP(B Wins)=[ANS]\nP(C Wins)=[ANS]", "answer_v3": [ "0.72", "0.24", "0.04" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0074", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability" ], "problem_v1": "Four candidates are running for mayor. The four candidates are Adams, Brown, Collins, and Dalton. Employing the subjective approach, a political scientist has assigned the following probabilitites: P(Adams wins)=0.53 P(Brown wins)=0.08 P(Collins wins)=0.25 P(Dalton wins)=0.14 Determine the probabilities of the following events: Adams loses=[ANS]\nEither Brown or Dalton wins=[ANS]\nEither Adams, Brown, or Collins wins=[ANS]", "answer_v1": [ "0.47", "0.22", "0.86" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Four candidates are running for mayor. The four candidates are Adams, Brown, Collins, and Dalton. Employing the subjective approach, a political scientist has assigned the following probabilitites: P(Adams wins)=0.32 P(Brown wins)=0.1 P(Collins wins)=0.21 P(Dalton wins)=0.37 Determine the probabilities of the following events: Adams loses=[ANS]\nEither Brown or Dalton wins=[ANS]\nEither Adams, Brown, or Collins wins=[ANS]", "answer_v2": [ "0.68", "0.47", "0.63" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Four candidates are running for mayor. The four candidates are Adams, Brown, Collins, and Dalton. Employing the subjective approach, a political scientist has assigned the following probabilitites: P(Adams wins)=0.39 P(Brown wins)=0.08 P(Collins wins)=0.22 P(Dalton wins)=0.31 Determine the probabilities of the following events: Adams loses=[ANS]\nEither Brown or Dalton wins=[ANS]\nEither Adams, Brown, or Collins wins=[ANS]", "answer_v3": [ "0.61", "0.39", "0.69" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0075", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "The following table lists the joint probabilities associated with smoking and lung disease among 60-to-65 year-old men.\n\\begin{array}{c|c|c} & \\mbox{Smoker} & \\mbox{Nonsmoker} \\\\ \\hline \\mbox{Has Lung Disease} & 0.11 & 0.04 \\\\ \\hline \\mbox{No Lung Disease} & 0.18 & 0.67 \\end{array} One 60-to-65 year old man is selected at random. What is the probability of the following events? A. He is a smoker: [ANS]\nB. He does not have lung disease: [ANS]\nC. He has lung disease given that he is a smoker: [ANS]\nD. He has lung disease given that he does not smoke: [ANS]", "answer_v1": [ "0.29", "0.85", "0.379310344827586", "0.0563380281690141" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The following table lists the joint probabilities associated with smoking and lung disease among 60-to-65 year-old men.\n\\begin{array}{c|c|c} & \\mbox{Smoker} & \\mbox{Nonsmoker} \\\\ \\hline \\mbox{Has Lung Disease} & 0.08 & 0.03 \\\\ \\hline \\mbox{No Lung Disease} & 0.2 & 0.69 \\end{array} One 60-to-65 year old man is selected at random. What is the probability of the following events? A. He is a smoker: [ANS]\nB. He does not have lung disease: [ANS]\nC. He has lung disease given that he is a smoker: [ANS]\nD. He has lung disease given that he does not smoke: [ANS]", "answer_v2": [ "0.28", "0.89", "0.285714285714286", "0.0416666666666667" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The following table lists the joint probabilities associated with smoking and lung disease among 60-to-65 year-old men.\n\\begin{array}{c|c|c} & \\mbox{Smoker} & \\mbox{Nonsmoker} \\\\ \\hline \\mbox{Has Lung Disease} & 0.09 & 0.03 \\\\ \\hline \\mbox{No Lung Disease} & 0.18 & 0.7 \\end{array} One 60-to-65 year old man is selected at random. What is the probability of the following events? A. He is a smoker: [ANS]\nB. He does not have lung disease: [ANS]\nC. He has lung disease given that he is a smoker: [ANS]\nD. He has lung disease given that he does not smoke: [ANS]", "answer_v3": [ "0.27", "0.88", "0.333333333333333", "0.0410958904109589" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0076", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "A department store analyzed its most recent sales and determined the relationship between the way a customer paid for an item and the price category of the item. The joint probabilities are given in the table below.\n$\\begin{array}{cccc}\\hline & Cash & Credit Card & Debit Card \\\\ \\hline Under 20 Dollars & 0.11 & 0.04 & 0.04 \\\\ \\hline 20-100 Dollars & 0.05 & 0.2 & 0.18 \\\\ \\hline Over 100 Dollars & 0.02 & 0.24 & 0.12 \\\\ \\hline \\end{array}$\nA. What proportion of purchases were paid with a debit card? Proportion=[ANS]\nB. Find the probability that a credit card purchase was for over 100 dollars. Probability=[ANS]\nC. Determine the proportion of purchases made by credit card or debit card. Proportion=[ANS]", "answer_v1": [ "0.34", "0.5", "0.82" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A department store analyzed its most recent sales and determined the relationship between the way a customer paid for an item and the price category of the item. The joint probabilities are given in the table below.\n$\\begin{array}{cccc}\\hline & Cash & Credit Card & Debit Card \\\\ \\hline Under 20 Dollars & 0.08 & 0.03 & 0.03 \\\\ \\hline 20-100 Dollars & 0.05 & 0.21 & 0.17 \\\\ \\hline Over 100 Dollars & 0.01 & 0.24 & 0.18 \\\\ \\hline \\end{array}$\nA. What proportion of purchases were paid with a debit card? Proportion=[ANS]\nB. Find the probability that a credit card purchase was for over 100 dollars. Probability=[ANS]\nC. Determine the proportion of purchases made by credit card or debit card. Proportion=[ANS]", "answer_v2": [ "0.38", "0.5", "0.86" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A department store analyzed its most recent sales and determined the relationship between the way a customer paid for an item and the price category of the item. The joint probabilities are given in the table below.\n$\\begin{array}{cccc}\\hline & Cash & Credit Card & Debit Card \\\\ \\hline Under 20 Dollars & 0.09 & 0.03 & 0.05 \\\\ \\hline 20-100 Dollars & 0.05 & 0.2 & 0.2 \\\\ \\hline Over 100 Dollars & 0.01 & 0.25 & 0.12 \\\\ \\hline \\end{array}$\nA. What proportion of purchases were paid with a debit card? Proportion=[ANS]\nB. Find the probability that a credit card purchase was for over 100 dollars. Probability=[ANS]\nC. Determine the proportion of purchases made by credit card or debit card. Proportion=[ANS]", "answer_v3": [ "0.37", "0.520833333333333", "0.85" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0077", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "The owner of an appliance store is interested in the relationship between the price at which an item is sold (regular or sale price) and the customer's decision on whether to purchase and extended warranty. After analyzing her records, she produced the following joint probabilities: \\begin{array}{c|c|c} & \\mbox{Purchased} & \\mbox{Did not purchase} \\\\ & \\mbox{extended warranty} & \\mbox{extended warranty} \\\\ \\hline \\mbox{Regular Price} & 0.23 & 0.56 \\\\ \\hline \\mbox{Sale Price} & 0.12 & 0.0899999999999999 \\\\ \\end{array} A. What is the probability that a customer who bought an item at the regular price purchased the extended warranty? Probability=[ANS]\nB. What is the probability that a customer buys an extended warranty? Probability=[ANS]", "answer_v1": [ "0.291139240506329", "0.35" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The owner of an appliance store is interested in the relationship between the price at which an item is sold (regular or sale price) and the customer's decision on whether to purchase and extended warranty. After analyzing her records, she produced the following joint probabilities: \\begin{array}{c|c|c} & \\mbox{Purchased} & \\mbox{Did not purchase} \\\\ & \\mbox{extended warranty} & \\mbox{extended warranty} \\\\ \\hline \\mbox{Regular Price} & 0.2 & 0.51 \\\\ \\hline \\mbox{Sale Price} & 0.14 & 0.15 \\\\ \\end{array} A. What is the probability that a customer who bought an item at the regular price purchased the extended warranty? Probability=[ANS]\nB. What is the probability that a customer buys an extended warranty? Probability=[ANS]", "answer_v2": [ "0.28169014084507", "0.34" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The owner of an appliance store is interested in the relationship between the price at which an item is sold (regular or sale price) and the customer's decision on whether to purchase and extended warranty. After analyzing her records, she produced the following joint probabilities: \\begin{array}{c|c|c} & \\mbox{Purchased} & \\mbox{Did not purchase} \\\\ & \\mbox{extended warranty} & \\mbox{extended warranty} \\\\ \\hline \\mbox{Regular Price} & 0.21 & 0.52 \\\\ \\hline \\mbox{Sale Price} & 0.13 & 0.14 \\\\ \\end{array} A. What is the probability that a customer who bought an item at the regular price purchased the extended warranty? Probability=[ANS]\nB. What is the probability that a customer buys an extended warranty? Probability=[ANS]", "answer_v3": [ "0.287671232876712", "0.34" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0078", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "A financial analyst has determined that there is a 28\\% probability that a mutual fund will outperform the market over a 1-year period provided that it outperforms the market the previous year. If only 16\\% of mutual funds outperform the market during any year, what is the probability that a mutual fund will outperform the market 2 years in a row? Probability=[ANS]", "answer_v1": [ "0.0448" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A financial analyst has determined that there is a 20\\% probability that a mutual fund will outperform the market over a 1-year period provided that it outperforms the market the previous year. If only 20\\% of mutual funds outperform the market during any year, what is the probability that a mutual fund will outperform the market 2 years in a row? Probability=[ANS]", "answer_v2": [ "0.04" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A financial analyst has determined that there is a 23\\% probability that a mutual fund will outperform the market over a 1-year period provided that it outperforms the market the previous year. If only 16\\% of mutual funds outperform the market during any year, what is the probability that a mutual fund will outperform the market 2 years in a row? Probability=[ANS]", "answer_v3": [ "0.0368" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0079", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "A foreman for an injection-molding firm admits that on 44\\% of his shifts, he forgets to shut off the injection machine on his line. This causes the machine to overheat, increasing the probability that a defective molding will be produced during the early morning run from 4\\% to 21\\%. If a molding is randomly selected from the early morning run of a random day, what is the probability that it is defective? Probability=[ANS]", "answer_v1": [ "0.1148" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A foreman for an injection-molding firm admits that on 13\\% of his shifts, he forgets to shut off the injection machine on his line. This causes the machine to overheat, increasing the probability that a defective molding will be produced during the early morning run from 5\\% to 16\\%. If a molding is randomly selected from the early morning run of a random day, what is the probability that it is defective? Probability=[ANS]", "answer_v2": [ "0.0643" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A foreman for an injection-molding firm admits that on 24\\% of his shifts, he forgets to shut off the injection machine on his line. This causes the machine to overheat, increasing the probability that a defective molding will be produced during the early morning run from 4\\% to 18\\%. If a molding is randomly selected from the early morning run of a random day, what is the probability that it is defective? Probability=[ANS]", "answer_v3": [ "0.0736" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0080", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "Determine all joint probabilities listed below from the following information: $P(A)=0.85, \\; P(A^c)=0.15, \\; P(B|A)=0.48, \\; P(B|A^c)=0.73$\n$P(A \\mbox{and} B)$=[ANS]\n$P(A \\mbox{and} B^c)$=[ANS]\n$P(A^c \\mbox{and} B)$=[ANS]\n$P(A^c \\mbox{and} B^c)$=[ANS]", "answer_v1": [ "0.408", "0.442", "0.1095", "0.0405" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Determine all joint probabilities listed below from the following information: $P(A)=0.71, \\; P(A^c)=0.29, \\; P(B|A)=0.58, \\; P(B|A^c)=0.63$\n$P(A \\mbox{and} B)$=[ANS]\n$P(A \\mbox{and} B^c)$=[ANS]\n$P(A^c \\mbox{and} B)$=[ANS]\n$P(A^c \\mbox{and} B^c)$=[ANS]", "answer_v2": [ "0.4118", "0.2982", "0.1827", "0.1073" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Determine all joint probabilities listed below from the following information: $P(A)=0.76, \\; P(A^c)=0.24, \\; P(B|A)=0.48, \\; P(B|A^c)=0.65$\n$P(A \\mbox{and} B)$=[ANS]\n$P(A \\mbox{and} B^c)$=[ANS]\n$P(A^c \\mbox{and} B)$=[ANS]\n$P(A^c \\mbox{and} B^c)$=[ANS]", "answer_v3": [ "0.3648", "0.3952", "0.156", "0.084" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0081", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "Approximately 13\\% of people are left-handed. If two people are selected at random, what is the probability of the following events? A. P(Both are right-handed)=[ANS]\nB. P(Both are left-handed)=[ANS]\nC. P(One is right-handed and the other is left-handed)=[ANS]\nD. P(At least one is right-handed)=[ANS]", "answer_v1": [ "0.7569", "0.0169", "0.2262", "0.9831" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Approximately 5\\% of people are left-handed. If two people are selected at random, what is the probability of the following events? A. P(Both are right-handed)=[ANS]\nB. P(Both are left-handed)=[ANS]\nC. P(One is right-handed and the other is left-handed)=[ANS]\nD. P(At least one is right-handed)=[ANS]", "answer_v2": [ "0.9025", "0.0025", "0.095", "0.9975" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Approximately 8\\% of people are left-handed. If two people are selected at random, what is the probability of the following events? A. P(Both are right-handed)=[ANS]\nB. P(Both are left-handed)=[ANS]\nC. P(One is right-handed and the other is left-handed)=[ANS]\nD. P(At least one is right-handed)=[ANS]", "answer_v3": [ "0.8464", "0.0064", "0.1472", "0.9936" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0082", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "An aerospace company has submitted bids on two separate federal government defense contracts. The company president believes that there is a 43\\% probability of winning the first contract. If they win the first contract, the probability of winning the second is 71\\%. However, if they lose the first contract, the president thinks that the probability of winning the second contract decreases to 51\\%. A. What is the probability that they win both contracts? Probability=[ANS]\nB. What is the probability that they lose both contracts? Probability=[ANS]\nC. What is the probability that they win only one contract? Probability=[ANS]", "answer_v1": [ "0.3053", "0.2793", "0.4154" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An aerospace company has submitted bids on two separate federal government defense contracts. The company president believes that there is a 35\\% probability of winning the first contract. If they win the first contract, the probability of winning the second is 75\\%. However, if they lose the first contract, the president thinks that the probability of winning the second contract decreases to 46\\%. A. What is the probability that they win both contracts? Probability=[ANS]\nB. What is the probability that they lose both contracts? Probability=[ANS]\nC. What is the probability that they win only one contract? Probability=[ANS]", "answer_v2": [ "0.2625", "0.351", "0.3865" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An aerospace company has submitted bids on two separate federal government defense contracts. The company president believes that there is a 38\\% probability of winning the first contract. If they win the first contract, the probability of winning the second is 71\\%. However, if they lose the first contract, the president thinks that the probability of winning the second contract decreases to 48\\%. A. What is the probability that they win both contracts? Probability=[ANS]\nB. What is the probability that they lose both contracts? Probability=[ANS]\nC. What is the probability that they win only one contract? Probability=[ANS]", "answer_v3": [ "0.2698", "0.3224", "0.4078" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0083", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "Researchers at the University of Pennsylvania School of Medicine have determined that children under 2 years old who sleep with the lights on have a 38\\% chance of becoming myopic before they are 16. Children who sleep in darkness have a 26\\% probability of becoming myopic. A survey indicates that 26\\% of children under 2 sleep with some light on. Find the probability that a random child under 2 will become myopic before reaching 16 years old. Probability=[ANS]", "answer_v1": [ "0.2912" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Researchers at the University of Pennsylvania School of Medicine have determined that children under 2 years old who sleep with the lights on have a 30\\% chance of becoming myopic before they are 16. Children who sleep in darkness have a 30\\% probability of becoming myopic. A survey indicates that 21\\% of children under 2 sleep with some light on. Find the probability that a random child under 2 will become myopic before reaching 16 years old. Probability=[ANS]", "answer_v2": [ "0.3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Researchers at the University of Pennsylvania School of Medicine have determined that children under 2 years old who sleep with the lights on have a 33\\% chance of becoming myopic before they are 16. Children who sleep in darkness have a 26\\% probability of becoming myopic. A survey indicates that 23\\% of children under 2 sleep with some light on. Find the probability that a random child under 2 will become myopic before reaching 16 years old. Probability=[ANS]", "answer_v3": [ "0.2761" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0084", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "A telemarketer calls people and tries to sell them a subscription to a daily newspaper. On 23\\% of her calls, there is no answer or the line is busy. She sells subscriptions to 8\\% of the remaining calls. For what proportion of calls does she make a sale? Proportion=[ANS]", "answer_v1": [ "0.0616" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A telemarketer calls people and tries to sell them a subscription to a daily newspaper. On 15\\% of her calls, there is no answer or the line is busy. She sells subscriptions to 10\\% of the remaining calls. For what proportion of calls does she make a sale? Proportion=[ANS]", "answer_v2": [ "0.085" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A telemarketer calls people and tries to sell them a subscription to a daily newspaper. On 18\\% of her calls, there is no answer or the line is busy. She sells subscriptions to 8\\% of the remaining calls. For what proportion of calls does she make a sale? Proportion=[ANS]", "answer_v3": [ "0.0656" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0085", "subject": "Probability", "topic": "Sample Space", "subtopic": "Probability: direct computation, inclusion/exclusion", "level": "1", "keywords": [ "probability", "conditional" ], "problem_v1": "In early 2001, the United States Census Bureau started releasing the results of the latest census. Among many other pieces of information, the bureau recoded the race or ethnicity of the residents of every county in every state. From these results the bureau calculated a 'diversity index,' which measures the probability that two people chosen at random are of different races or ethnicities. The census determined that in a county in Wisconsin, 83\\% of its residents are white, 9\\% are black, and 8\\% are asian. Calculate the diversity index for this county. Diversity Index=[ANS]", "answer_v1": [ "0.2966" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In early 2001, the United States Census Bureau started releasing the results of the latest census. Among many other pieces of information, the bureau recoded the race or ethnicity of the residents of every county in every state. From these results the bureau calculated a 'diversity index,' which measures the probability that two people chosen at random are of different races or ethnicities. The census determined that in a county in Wisconsin, 75\\% of its residents are white, 10\\% are black, and 15\\% are asian. Calculate the diversity index for this county. Diversity Index=[ANS]", "answer_v2": [ "0.405" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In early 2001, the United States Census Bureau started releasing the results of the latest census. Among many other pieces of information, the bureau recoded the race or ethnicity of the residents of every county in every state. From these results the bureau calculated a 'diversity index,' which measures the probability that two people chosen at random are of different races or ethnicities. The census determined that in a county in Wisconsin, 78\\% of its residents are white, 9\\% are black, and 13\\% are asian. Calculate the diversity index for this county. Diversity Index=[ANS]", "answer_v3": [ "0.3666" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0086", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "Dice", "Probability" ], "problem_v1": "Two six-sided dice are rolled (one red one and one green one). Some possibilities are (Red=1,Green=5) or (Red=2,Green=2) etc.\n(a) How many total possibilities are there? [ANS]\nFor the rest of the questions, we will assume that the dice are fair and that all of the possibilities in (a) are equally likely.\n(b) What is the probability that the sum on the two dice comes out to be 11? [ANS]\n(c) What is the probability that the sum on the two dice comes out to be 10? Answer: [ANS]\n(d) What is the probability that the numbers on the two dice are equal? Answer: [ANS]", "answer_v1": [ "36", "0.0555555555555556", "0.0833333333333333", "0.166666666666667" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Two six-sided dice are rolled (one red one and one green one). Some possibilities are (Red=1,Green=5) or (Red=2,Green=2) etc.\n(a) How many total possibilities are there? [ANS]\nFor the rest of the questions, we will assume that the dice are fair and that all of the possibilities in (a) are equally likely.\n(b) What is the probability that the sum on the two dice comes out to be 7? [ANS]\n(c) What is the probability that the sum on the two dice comes out to be 12? Answer: [ANS]\n(d) What is the probability that the numbers on the two dice are equal? Answer: [ANS]", "answer_v2": [ "36", "0.166666666666667", "0.0277777777777778", "0.166666666666667" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Two six-sided dice are rolled (one red one and one green one). Some possibilities are (Red=1,Green=5) or (Red=2,Green=2) etc.\n(a) How many total possibilities are there? [ANS]\nFor the rest of the questions, we will assume that the dice are fair and that all of the possibilities in (a) are equally likely.\n(b) What is the probability that the sum on the two dice comes out to be 8? [ANS]\n(c) What is the probability that the sum on the two dice comes out to be 10? Answer: [ANS]\n(d) What is the probability that the numbers on the two dice are equal? Answer: [ANS]", "answer_v3": [ "36", "0.138888888888889", "0.0833333333333333", "0.166666666666667" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0087", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "4", "keywords": [ "Probability", "Discrete" ], "problem_v1": "In the game Roulette, a ball spins on a circular wheel that is divided into 38 arcs of equal lenght, numbered 00, 0, 1, 2,..., 35, 36. The number on the arc on which the ball stops is the outcome of one play of the game. The numbers are also colored as follows: 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 are red, 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 are black, 0, 00 are green Define the following events: $A: \\{$ Outcome is an even number (0 and 00 are considered neither odd nor even) $\\}$ $B: \\{$ Outcome is a red number $\\}$ $C: \\{$ Outcome is a green number $\\}$ $D: \\{$ Outcome is a low number (1-18) $\\}$ Find the following probablilities:\n(a) $P(D)$ $=$ [ANS]\n(b) $P(A \\cap B)$ $=$ [ANS]\n(c) $P(A \\cup C \\cup D)$ $=$ [ANS]", "answer_v1": [ "0.473684210526316", "0.210526315789474", "0.763157894736842" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In the game Roulette, a ball spins on a circular wheel that is divided into 38 arcs of equal lenght, numbered 00, 0, 1, 2,..., 35, 36. The number on the arc on which the ball stops is the outcome of one play of the game. The numbers are also colored as follows: 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 are red, 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 are black, 0, 00 are green Define the following events: $A: \\{$ Outcome is an even number (0 and 00 are considered neither odd nor even) $\\}$ $B: \\{$ Outcome is a red number $\\}$ $C: \\{$ Outcome is a green number $\\}$ $D: \\{$ Outcome is a low number (1-18) $\\}$ Find the following probablilities:\n(a) $P(A)$ $=$ [ANS]\n(b) $P(C \\cap D)$ $=$ [ANS]\n(c) $P(A \\cup B \\cup C)$ $=$ [ANS]", "answer_v2": [ "0.473684210526316", "0", "0.789473684210526" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In the game Roulette, a ball spins on a circular wheel that is divided into 38 arcs of equal lenght, numbered 00, 0, 1, 2,..., 35, 36. The number on the arc on which the ball stops is the outcome of one play of the game. The numbers are also colored as follows: 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 are red, 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 are black, 0, 00 are green Define the following events: $A: \\{$ Outcome is an even number (0 and 00 are considered neither odd nor even) $\\}$ $B: \\{$ Outcome is a red number $\\}$ $C: \\{$ Outcome is a green number $\\}$ $D: \\{$ Outcome is a low number (1-18) $\\}$ Find the following probablilities:\n(a) $P(B)$ $=$ [ANS]\n(b) $P(A \\cap C)$ $=$ [ANS]\n(c) $P(A \\cup B \\cup D)$ $=$ [ANS]", "answer_v3": [ "0.473684210526316", "0", "0.842105263157895" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0088", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "Probability", "Discrete" ], "problem_v1": "In a study by the Department of Transportation, there were a total of 95 drivers that were pulled over for speeding. Out of those 95 drivers, 36 were men who were ticketed, 12 were men who were not ticketed, 8 were women who were ticketed, and 39 were women who were not ticketed. Suppose one person was chosen at random.\n(a) What is the probability that the selected person is a man or someone who was not ticketed? Answer: [ANS]\n(b) What is the probability that the selected person is a woman or someone who was not ticketed? Answer: [ANS]", "answer_v1": [ "0.915789473684211", "0.621052631578948" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "In a study by the Department of Transportation, there were a total of 81 drivers that were pulled over for speeding. Out of those 81 drivers, 40 were men who were ticketed, 9 were men who were not ticketed, 5 were women who were ticketed, and 27 were women who were not ticketed. Suppose one person was chosen at random.\n(a) What is the probability that the selected person is a woman who was not ticketed? Answer: [ANS]\n(b) What is the probability that the selected person is a man or someone who was not ticketed? Answer: [ANS]", "answer_v2": [ "0.333333333333333", "0.938271604938272" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "In a study by the Department of Transportation, there were a total of 86 drivers that were pulled over for speeding. Out of those 86 drivers, 36 were men who were ticketed, 10 were men who were not ticketed, 7 were women who were ticketed, and 33 were women who were not ticketed. Suppose one person was chosen at random.\n(a) What is the probability that the selected person is a woman or someone who was ticketed? Answer: [ANS]\n(b) What is the probability that the selected person is a woman or someone who was not ticketed? Answer: [ANS]", "answer_v3": [ "0.883720930232558", "0.581395348837209" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0089", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "4", "keywords": [ "Probability", "Discrete" ], "problem_v1": "The sample space for an experiment contains five sample points. The probabilities of the sample points are:\n$P(1)=P(2)=0.2$ $P(3)=P(4)=0.15$ $P(5)=0.3$ Find the probability of each of the following events:\n$A: \\ \\{$ Either 4 or 3 occurs $\\}$ $B: \\ \\{$ Either 4, 1, or 2 occurs $\\}$ $C: \\ \\{$ 5 does not occur $\\}$ $P(A)=$ [ANS]\n$P(B)=$ [ANS]\n$P(C)=$ [ANS]", "answer_v1": [ "0.3", "0.55", "0.7" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The sample space for an experiment contains five sample points. The probabilities of the sample points are:\n$P(1)=P(2)=0.05$ $P(3)=P(4)=0.2$ $P(5)=0.5$ Find the probability of each of the following events:\n$A: \\ \\{$ Either 1 or 3 occurs $\\}$ $B: \\ \\{$ Either 1, 5, or 2 occurs $\\}$ $C: \\ \\{$ 4 does not occur $\\}$ $P(A)=$ [ANS]\n$P(B)=$ [ANS]\n$P(C)=$ [ANS]", "answer_v2": [ "0.25", "0.6", "0.8" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The sample space for an experiment contains five sample points. The probabilities of the sample points are:\n$P(1)=P(2)=0.1$ $P(3)=P(4)=0.15$ $P(5)=0.5$ Find the probability of each of the following events:\n$A: \\ \\{$ Either 2 or 4 occurs $\\}$ $B: \\ \\{$ Either 2, 1, or 3 occurs $\\}$ $C: \\ \\{$ 5 does not occur $\\}$ $P(A)=$ [ANS]\n$P(B)=$ [ANS]\n$P(C)=$ [ANS]", "answer_v3": [ "0.25", "0.35", "0.5" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0090", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "Conditional", "Probability" ], "problem_v1": "In a certain community, 35\\% of the families own a dog, and 20\\% of the families that own a dog also own a cat. It is also known that 33\\% of all the families own a cat. What is the probability that a randomly selected family owns a dog? [ANS]\nWhat is the conditional probability that a randomly selected family owns a dog given that it owns a cat? [ANS]", "answer_v1": [ "0.35", "0.212121212121212" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "In a certain community, 20\\% of the families own a dog, and 20\\% of the families that own a dog also own a cat. It is also known that 25\\% of all the families own a cat. What is the probability that a randomly selected family owns a cat? [ANS]\nWhat is the conditional probability that a randomly selected family doesn't own a dog given that it owns a cat? [ANS]", "answer_v2": [ "0.25", "0.84" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "In a certain community, 25\\% of the families own a dog, and 20\\% of the families that own a dog also own a cat. It is also known that 28\\% of all the families own a cat. What is the probability that a randomly selected family owns a cat? [ANS]\nWhat is the conditional probability that a randomly selected family owns a dog given that it owns a cat? [ANS]", "answer_v3": [ "0.28", "0.178571428571429" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0091", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "Conditional", "Probability" ], "problem_v1": "A box contains one yellow, two red, and three green balls. Two balls are randomly chosen without replacement. Define the following events: $A: \\lbrace$ One of the balls is yellow $\\rbrace$ $B: \\lbrace$ At least one ball is red $\\rbrace$ $C: \\lbrace$ Both balls are green $\\rbrace$ $D: \\lbrace$ Both balls are of the same color $\\rbrace$ Find the following conditional probabilities:\n(a) $P(A|\\overline{B})$ $=$ [ANS]\n(b) $P(D|\\overline{B})$ $=$ [ANS]\n(c) $P(C|A)$ $=$ [ANS]", "answer_v1": [ "0.5", "0.5", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A box contains one yellow, two red, and three green balls. Two balls are randomly chosen without replacement. Define the following events: $A: \\lbrace$ One of the balls is yellow $\\rbrace$ $B: \\lbrace$ At least one ball is red $\\rbrace$ $C: \\lbrace$ Both balls are green $\\rbrace$ $D: \\lbrace$ Both balls are of the same color $\\rbrace$ Find the following conditional probabilities:\n(a) $P(A|B)$ $=$ [ANS]\n(b) $P(\\overline{D}|B)$ $=$ [ANS]\n(c) $P(C|D)$ $=$ [ANS]", "answer_v2": [ "0.222222", "0.888889", "0.75" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A box contains one yellow, two red, and three green balls. Two balls are randomly chosen without replacement. Define the following events: $A: \\lbrace$ One of the balls is yellow $\\rbrace$ $B: \\lbrace$ At least one ball is red $\\rbrace$ $C: \\lbrace$ Both balls are green $\\rbrace$ $D: \\lbrace$ Both balls are of the same color $\\rbrace$ Find the following conditional probabilities:\n(a) $P(B|A)$ $=$ [ANS]\n(b) $P(D|\\overline{B})$ $=$ [ANS]\n(c) $P(D|C)$ $=$ [ANS]", "answer_v3": [ "0.4", "0.5", "1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0092", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "Conditional", "Probability" ], "problem_v1": "\"Channel One\" is an educational television network for which participating secondary schools are equipped with TV sets in every classroom. It has been found that $70$ \\% of secondary schools subscribe to Channel One, where of these subscribers $15$ \\% never use Channel One while $30$ \\% claim to use it more than 5 times per week. Find the probability that a randomly selected seconday school subscribes to Channel One and uses it more than 5 times per week. answer: [ANS]", "answer_v1": [ "0.21" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "\"Channel One\" is an educational television network for which participating secondary schools are equipped with TV sets in every classroom. It has been found that $30$ \\% of secondary schools subscribe to Channel One, where of these subscribers $20$ \\% never use Channel One while $15$ \\% claim to use it more than 5 times per week. Find the probability that a randomly selected seconday school subscribes to Channel One but never uses it. answer: [ANS]", "answer_v2": [ "0.06" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "\"Channel One\" is an educational television network for which participating secondary schools are equipped with TV sets in every classroom. It has been found that $45$ \\% of secondary schools subscribe to Channel One, where of these subscribers $15$ \\% never use Channel One while $15$ \\% claim to use it more than 5 times per week. Find the probability that a randomly selected seconday school subscribes to Channel One and uses it more than 5 times per week. answer: [ANS]", "answer_v3": [ "0.0675" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0093", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "In a survey of 298 people, the following data were obtained relating gender to political orientation: $\\begin{array}{ccccc}\\hline & Republican (R) & Democrat (D) & Independent (I) & Total \\\\ \\hline Male (M) & 88 & 55 & 29 & 172 \\\\ \\hline Femal (F) & 75 & 38 & 13 & 126 \\\\ \\hline Total & 163 & 93 & 42 & 298 \\\\ \\hline \\end{array}$\nA person is randomly selected. What is the probability that the person is: a) Male? [ANS]\nb) Male and a Democrat? [ANS]\nc) Male given that the person is a Democrat? [ANS]\nd) Republican given that the person is Male? [ANS]\ne) Female given that the person is an Independent? [ANS]\nf) Are the events Male and Republican independent? [ANS] Enter yes or no.", "answer_v1": [ "0.577181208053691", "0.184563758389262", "0.591397849462366", "0.511627906976744", "0.30952380952381", "no" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "In a survey of 274 people, the following data were obtained relating gender to political orientation: $\\begin{array}{ccccc}\\hline & Republican (R) & Democrat (D) & Independent (I) & Total \\\\ \\hline Male (M) & 54 & 76 & 14 & 144 \\\\ \\hline Femal (F) & 40 & 77 & 13 & 130 \\\\ \\hline Total & 94 & 153 & 27 & 274 \\\\ \\hline \\end{array}$\nA person is randomly selected. What is the probability that the person is: a) Male? [ANS]\nb) Male and a Democrat? [ANS]\nc) Male given that the person is a Democrat? [ANS]\nd) Republican given that the person is Male? [ANS]\ne) Female given that the person is an Independent? [ANS]\nf) Are the events Male and Republican independent? [ANS] Enter yes or no.", "answer_v2": [ "0.525547445255474", "0.277372262773723", "0.496732026143791", "0.375", "0.481481481481481", "no" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "In a survey of 243 people, the following data were obtained relating gender to political orientation: $\\begin{array}{ccccc}\\hline & Republican (R) & Democrat (D) & Independent (I) & Total \\\\ \\hline Male (M) & 65 & 56 & 18 & 139 \\\\ \\hline Femal (F) & 59 & 32 & 13 & 104 \\\\ \\hline Total & 124 & 88 & 31 & 243 \\\\ \\hline \\end{array}$\nA person is randomly selected. What is the probability that the person is: a) Male? [ANS]\nb) Male and a Democrat? [ANS]\nc) Male given that the person is a Democrat? [ANS]\nd) Republican given that the person is Male? [ANS]\ne) Female given that the person is an Independent? [ANS]\nf) Are the events Male and Republican independent? [ANS] Enter yes or no.", "answer_v3": [ "0.57201646090535", "0.230452674897119", "0.636363636363636", "0.467625899280576", "0.419354838709677", "no" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0094", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "A box contains 90 balls numbered from 1 to 90. If 11 balls are drawn with replacement, what is the probability that at least two of them have the same number? Answer: [ANS]", "answer_v1": [ "0.470787064977858" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A box contains 50 balls numbered from 1 to 50. If 15 balls are drawn with replacement, what is the probability that at least two of them have the same number? Answer: [ANS]", "answer_v2": [ "0.903552237006987" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A box contains 65 balls numbered from 1 to 65. If 12 balls are drawn with replacement, what is the probability that at least two of them have the same number? Answer: [ANS]", "answer_v3": [ "0.660807912689776" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0095", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "If $P(E \\cap F)=0.21$, $P(E |F)=0.42$, and $P(F | E)=0.6$, then\n(a) $P(E)=$ [ANS]\n(b) $P(F)=$ [ANS]\n(c) $P(E \\cup F)=$ [ANS]", "answer_v1": [ "0.35", "0.5", "0.64" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If $P(E \\cap F)=0.018$, $P(E |F)=0.09$, and $P(F | E)=0.9$, then\n(a) $P(E)=$ [ANS]\n(b) $P(F)=$ [ANS]\n(c) $P(E \\cup F)=$ [ANS]", "answer_v2": [ "0.02", "0.2", "0.202" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If $P(E \\cap F)=0.054$, $P(E |F)=0.18$, and $P(F | E)=0.6$, then\n(a) $P(E)=$ [ANS]\n(b) $P(F)=$ [ANS]\n(c) $P(E \\cup F)=$ [ANS]", "answer_v3": [ "0.09", "0.3", "0.336" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0096", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "In a survey of 241 people, the following data were obtained relating gender to color-blindness: $\\begin{array}{cccc}\\hline & Color-Blind (C) & Not Color-Blind \\overline C & Total \\\\ \\hline Male (M) & 88 & 59 & 147 \\\\ \\hline Female (F) & 52 & 42 & 94 \\\\ \\hline Total & 140 & 101 & 241 \\\\ \\hline \\end{array}$\nA person is randomly selected. What is the probability that the person is: a) Male? [ANS]\nb) Male and Color-blind? [ANS]\nc) Male given that the person is Color-blind? [ANS]\nd) Color-blind given that the person is Male? [ANS]\ne) Female given that the person is not Color-blind? [ANS]\nf) Are the events Male and Color blind independent? [ANS] Enter yes or no.", "answer_v1": [ "0.609958506224066", "0.365145228215768", "0.628571428571429", "0.598639455782313", "0.415841584158416", "no" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "In a survey of 142 people, the following data were obtained relating gender to color-blindness: $\\begin{array}{cccc}\\hline & Color-Blind (C) & Not Color-Blind \\overline C & Total \\\\ \\hline Male (M) & 54 & 51 & 105 \\\\ \\hline Female (F) & 15 & 22 & 37 \\\\ \\hline Total & 69 & 73 & 142 \\\\ \\hline \\end{array}$\nA person is randomly selected. What is the probability that the person is: a) Male? [ANS]\nb) Male and Color-blind? [ANS]\nc) Male given that the person is Color-blind? [ANS]\nd) Color-blind given that the person is Male? [ANS]\ne) Female given that the person is not Color-blind? [ANS]\nf) Are the events Male and Color blind independent? [ANS] Enter yes or no.", "answer_v2": [ "0.73943661971831", "0.380281690140845", "0.782608695652174", "0.514285714285714", "0.301369863013699", "no" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "In a survey of 162 people, the following data were obtained relating gender to color-blindness: $\\begin{array}{cccc}\\hline & Color-Blind (C) & Not Color-Blind \\overline C & Total \\\\ \\hline Male (M) & 65 & 47 & 112 \\\\ \\hline Female (F) & 22 & 28 & 50 \\\\ \\hline Total & 87 & 75 & 162 \\\\ \\hline \\end{array}$\nA person is randomly selected. What is the probability that the person is: a) Male? [ANS]\nb) Male and Color-blind? [ANS]\nc) Male given that the person is Color-blind? [ANS]\nd) Color-blind given that the person is Male? [ANS]\ne) Female given that the person is not Color-blind? [ANS]\nf) Are the events Male and Color blind independent? [ANS] Enter yes or no.", "answer_v3": [ "0.691358024691358", "0.401234567901235", "0.747126436781609", "0.580357142857143", "0.373333333333333", "no" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0097", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "A box contains 23 yellow, 30 green and 37 red jelly beans. If 13 jelly beans are selected at random, what is the probability that: a) 5 are yellow? [ANS]\nb) 5 are yellow and 7 are green? [ANS]\nc) At least one is yellow? [ANS]", "answer_v1": [ "0.133548064996347", "0.00154230250563169", "0.98456128244257" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A box contains 15 yellow, 34 green and 35 red jelly beans. If 10 jelly beans are selected at random, what is the probability that: a) 8 are yellow? [ANS]\nb) 8 are yellow and 1 are green? [ANS]\nc) At least one is yellow? [ANS]", "answer_v2": [ "5.46771765802354E-06", "2.77347997146122E-06", "0.876845613490025" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A box contains 18 yellow, 29 green and 33 red jelly beans. If 12 jelly beans are selected at random, what is the probability that: a) 3 are yellow? [ANS]\nb) 3 are yellow and 8 are green? [ANS]\nc) At least one is yellow? [ANS]", "answer_v3": [ "0.274768146720895", "0.00191842855591494", "0.964144838429792" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0098", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability", "conditional" ], "problem_v1": "Events $A_1$, $A_2$ and $A_3$ form a partiton of the sample space S with probabilities $P(A_1)=0.4$, $P(A_2)=0.3$, $P(A_3)=0.3$. If E is an event in S with $P(E|A_1)=0.4$, $P(E|A_2)=0.5$, $P(E|A_3)=0.4$, compute\n$P(E)=$ [ANS]\n$P(A_1|E)=$ [ANS]\n$P(A_2|E)=$ [ANS]\n$P(A_3|E)=$ [ANS]", "answer_v1": [ "0.43", "0.372093023255814", "0.348837209302326", "0.27906976744186" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Events $A_1$, $A_2$ and $A_3$ form a partiton of the sample space S with probabilities $P(A_1)=0.1$, $P(A_2)=0.5$, $P(A_3)=0.4$. If E is an event in S with $P(E|A_1)=0.1$, $P(E|A_2)=0.3$, $P(E|A_3)=0.8$, compute\n$P(E)=$ [ANS]\n$P(A_1|E)=$ [ANS]\n$P(A_2|E)=$ [ANS]\n$P(A_3|E)=$ [ANS]", "answer_v2": [ "0.48", "0.0208333333333333", "0.3125", "0.666666666666667" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Events $A_1$, $A_2$ and $A_3$ form a partiton of the sample space S with probabilities $P(A_1)=0.2$, $P(A_2)=0.4$, $P(A_3)=0.4$. If E is an event in S with $P(E|A_1)=0.2$, $P(E|A_2)=0.4$, $P(E|A_3)=0.4$, compute\n$P(E)=$ [ANS]\n$P(A_1|E)=$ [ANS]\n$P(A_2|E)=$ [ANS]\n$P(A_3|E)=$ [ANS]", "answer_v3": [ "0.36", "0.111111111111111", "0.444444444444444", "0.444444444444444" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0099", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "sample space", "coin toss" ], "problem_v1": "In an experiment, a fair coin is tossed 13 times and the face that appears ($H$ for head or $T$ for tail) for each toss is recorded. How many elements of the sample space will have no tails? [ANS]\nHow many elements of the sample space will have exactly one tail? [ANS]\nHow many elements of the sample space will start with a pair ($TT$ or $HH$) or end with a pair (but not both) and have a total of exactly two tails? [ANS]\nHow many elements of the sample space will start or end with a tail and have an adjacent pair or triple of tails and include a total of exactly three tails? [ANS]", "answer_v1": [ "1", "13", "36", "40" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "In an experiment, a fair coin is tossed 7 times and the face that appears ($H$ for head or $T$ for tail) for each toss is recorded. How many elements of the sample space will have no tails? [ANS]\nHow many elements of the sample space will have exactly one tail? [ANS]\nHow many elements of the sample space will start or end (or both) with a tail and have a total of exactly two tails? [ANS]\nHow many elements of the sample space will start and end with a tail and include a total of exactly three tails? [ANS]", "answer_v2": [ "1", "7", "11", "5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "In an experiment, a fair coin is tossed 9 times and the face that appears ($H$ for head or $T$ for tail) for each toss is recorded. How many elements of the sample space will have no tails? [ANS]\nHow many elements of the sample space will have exactly one tail? [ANS]\nHow many elements of the sample space will start or end (or both) with a tail and have a total of exactly two tails? [ANS]\nHow many elements of the sample space will start or end with a tail and have an adjacent pair or triple of tails and include a total of exactly three tails? [ANS]", "answer_v3": [ "1", "9", "15", "24" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0100", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability" ], "problem_v1": "In the rolling of two fair dice calculate the following: P(Sum of the two dice is 9)=[ANS]. P(Sum of the two dice is 7)=[ANS]. P(Sum of the two dice is not 8)=[ANS]. P(Sum of the two dice is 6 or 4)=[ANS]. P(Sum of the two dice is not 5 and not 10)=[ANS].", "answer_v1": [ "0.111111111111111", "0.166666666666667", "0.861111111111111", "0.222222222222222", "0.805555555555556" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "In the rolling of two fair dice calculate the following: P(Sum of the two dice is 4)=[ANS]. P(Sum of the two dice is 10)=[ANS]. P(Sum of the two dice is not 5)=[ANS]. P(Sum of the two dice is 7 or 9)=[ANS]. P(Sum of the two dice is not 6 and not 8)=[ANS].", "answer_v2": [ "0.0833333333333333", "0.0833333333333333", "0.888888888888889", "0.277777777777778", "0.722222222222222" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "In the rolling of two fair dice calculate the following: P(Sum of the two dice is 6)=[ANS]. P(Sum of the two dice is 8)=[ANS]. P(Sum of the two dice is not 5)=[ANS]. P(Sum of the two dice is 9 or 4)=[ANS]. P(Sum of the two dice is not 7 and not 10)=[ANS].", "answer_v3": [ "0.138888888888889", "0.138888888888889", "0.888888888888889", "0.194444444444444", "0.75" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0101", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "Bayes", "probability", "conditional" ], "problem_v1": "You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.85. With water it will die with probability 0.5. You are 86 \\% certain the neighbor will remember to water the plant. When you are on vacation, find the probability that the plant will die. Answer: [ANS]\nYou come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it? Answer: [ANS]", "answer_v1": [ "0.549", "0.216757741347905" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.7. With water it will die with probability 0.6. You are 81 \\% certain the neighbor will remember to water the plant. When you are on vacation, find the probability that the plant will die. Answer: [ANS]\nYou come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it? Answer: [ANS]", "answer_v2": [ "0.619", "0.21486268174475" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.75. With water it will die with probability 0.55. You are 83 \\% certain the neighbor will remember to water the plant. When you are on vacation, find the probability that the plant will die. Answer: [ANS]\nYou come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it? Answer: [ANS]", "answer_v3": [ "0.584", "0.218321917808219" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0102", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability" ], "problem_v1": "Consider the experiment where a fair coin is tossed. If the toss results in a head, then 1 die is thrown. If the toss results in a tail, then 2 dice are thrown. Let X denote the random variable that counts the number of spots showing on the thrown die or dice. The range of values that X can assume are the positive integers from 1 to 12 inclusive. Please give the corresponding probabilities for the values of X given below. Pr(X=1)=[ANS]\nPr(X=2)=[ANS]\nPr(X=3)=[ANS]\nPr(X=4)=[ANS]\nPr(X=5)=[ANS]\nPr(X=6)=[ANS]\nPr(X=7)=[ANS]\nPr(X=8)=[ANS]\nPr(X=9)=[ANS]\nPr(X=10)=[ANS]\nPr(X=11)=[ANS]\nPr(X=12)=[ANS]\nFurther, find the probability that X is divisible by 5. Probability that X is divisible by 5 equals [ANS]", "answer_v1": [ "0.0833333333333333", "0.0972222222222222", "0.111111111111111", "0.125", "0.138888888888889", "0.152777777777778", "0.0833333333333333", "0.0694444444444444", "0.0555555555555556", "0.0416666666666667", "0.0277777777777778", "0.0138888888888889", "0.180555555555556" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the experiment where a fair coin is tossed. If the toss results in a head, then 1 die is thrown. If the toss results in a tail, then 2 dice are thrown. Let X denote the random variable that counts the number of spots showing on the thrown die or dice. The range of values that X can assume are the positive integers from 1 to 12 inclusive. Please give the corresponding probabilities for the values of X given below. Pr(X=1)=[ANS]\nPr(X=2)=[ANS]\nPr(X=3)=[ANS]\nPr(X=4)=[ANS]\nPr(X=5)=[ANS]\nPr(X=6)=[ANS]\nPr(X=7)=[ANS]\nPr(X=8)=[ANS]\nPr(X=9)=[ANS]\nPr(X=10)=[ANS]\nPr(X=11)=[ANS]\nPr(X=12)=[ANS]\nFurther, find the probability that X is divisible by 2. Probability that X is divisible by 2 equals [ANS]", "answer_v2": [ "0.0833333333333333", "0.0972222222222222", "0.111111111111111", "0.125", "0.138888888888889", "0.152777777777778", "0.0833333333333333", "0.0694444444444444", "0.0555555555555556", "0.0416666666666667", "0.0277777777777778", "0.0138888888888889", "0.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the experiment where a fair coin is tossed. If the toss results in a head, then 1 die is thrown. If the toss results in a tail, then 2 dice are thrown. Let X denote the random variable that counts the number of spots showing on the thrown die or dice. The range of values that X can assume are the positive integers from 1 to 12 inclusive. Please give the corresponding probabilities for the values of X given below. Pr(X=1)=[ANS]\nPr(X=2)=[ANS]\nPr(X=3)=[ANS]\nPr(X=4)=[ANS]\nPr(X=5)=[ANS]\nPr(X=6)=[ANS]\nPr(X=7)=[ANS]\nPr(X=8)=[ANS]\nPr(X=9)=[ANS]\nPr(X=10)=[ANS]\nPr(X=11)=[ANS]\nPr(X=12)=[ANS]\nFurther, find the probability that X is divisible by 3. Probability that X is divisible by 3 equals [ANS]", "answer_v3": [ "0.0833333333333333", "0.0972222222222222", "0.111111111111111", "0.125", "0.138888888888889", "0.152777777777778", "0.0833333333333333", "0.0694444444444444", "0.0555555555555556", "0.0416666666666667", "0.0277777777777778", "0.0138888888888889", "0.333333333333333" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Probability_0103", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability" ], "problem_v1": "Employment data at a large company reveal that 58 \\% of the workers are married, that 33 \\% are college graduates, and that 1/3 of the college graduates are married. What is the probability that a randomly chosen worker is: a) neither married nor a college graduate? Answer=[ANS] \\% b) married but not a college graduate? Answer=[ANS] \\% c) married or a college graduate? Answer=[ANS] \\%", "answer_v1": [ "20", "47", "80" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Employment data at a large company reveal that 50 \\% of the workers are married, that 43 \\% are college graduates, and that 1/6 of the college graduates are married. What is the probability that a randomly chosen worker is: a) neither married nor a college graduate? Answer=[ANS] \\% b) married but not a college graduate? Answer=[ANS] \\% c) married or a college graduate? Answer=[ANS] \\%", "answer_v2": [ "14.1666666666667", "42.8333333333333", "85.8333333333333" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Employment data at a large company reveal that 53 \\% of the workers are married, that 33 \\% are college graduates, and that 1/5 of the college graduates are married. What is the probability that a randomly chosen worker is: a) neither married nor a college graduate? Answer=[ANS] \\% b) married but not a college graduate? Answer=[ANS] \\% c) married or a college graduate? Answer=[ANS] \\%", "answer_v3": [ "20.6", "46.4", "79.4" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0104", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "Bayes", "probability", "conditional" ], "problem_v1": "A new medical test has been designed to detect the presence of the mysterious Brainlesserian disease. Among those who have the disease, the probability that the disease will be detected by the new test is 0.86. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.08. It is estimated that 16 \\% of the population who take this test have the disease. If the test administered to an individual is positive, what is the probability that the person actually has the disease? Answer: [ANS]", "answer_v1": [ "0.671875" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A new medical test has been designed to detect the presence of the mysterious Brainlesserian disease. Among those who have the disease, the probability that the disease will be detected by the new test is 0.7. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.12. It is estimated that 11 \\% of the population who take this test have the disease. If the test administered to an individual is positive, what is the probability that the person actually has the disease? Answer: [ANS]", "answer_v2": [ "0.418933623503809" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A new medical test has been designed to detect the presence of the mysterious Brainlesserian disease. Among those who have the disease, the probability that the disease will be detected by the new test is 0.74. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.08. It is estimated that 13 \\% of the population who take this test have the disease. If the test administered to an individual is positive, what is the probability that the person actually has the disease? Answer: [ANS]", "answer_v3": [ "0.58021712907117" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0105", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "Bayes", "probability", "conditional" ], "problem_v1": "Of 380 male and 220 female employees at the Flagstaff Mall, 260 of the men and 140 of the women are on flex-time (flexible working hours). Given that an employee selected at random from this group is on flex-time, what is the probability that the employee is a woman?\nAnswer: [ANS]", "answer_v1": [ "0.35" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Of 300 male and 300 female employees at the Flagstaff Mall, 200 of the men and 200 of the women are on flex-time (flexible working hours). Given that an employee selected at random from this group is on flex-time, what is the probability that the employee is a woman?\nAnswer: [ANS]", "answer_v2": [ "0.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Of 330 male and 270 female employees at the Flagstaff Mall, 210 of the men and 180 of the women are on flex-time (flexible working hours). Given that an employee selected at random from this group is on flex-time, what is the probability that the employee is a woman?\nAnswer: [ANS]", "answer_v3": [ "0.461538461538462" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0106", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "Bayes", "probability", "conditional" ], "problem_v1": "Factories A, B and C produce computers. Factory A produces 4 times as many computers as factory C, and factory B produces 6 times as many computers as factory C. The probability that a computer produced by factory A is defective is 0.029, the probability that a computer produced by factory B is defective is 0.042, and the probability that a computer produced by factory C is defective is 0.039. A computer is selected at random and it is found to be defective. What is the probability it came from factory A? Answer: [ANS]", "answer_v1": [ "0.285012285012285" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Factories A, B and C produce computers. Factory A produces 2 times as many computers as factory C, and factory B produces 7 times as many computers as factory C. The probability that a computer produced by factory A is defective is 0.014, the probability that a computer produced by factory B is defective is 0.03, and the probability that a computer produced by factory C is defective is 0.059. A computer is selected at random and it is found to be defective. What is the probability it came from factory A? Answer: [ANS]", "answer_v2": [ "0.0942760942760943" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Factories A, B and C produce computers. Factory A produces 2 times as many computers as factory C, and factory B produces 6 times as many computers as factory C. The probability that a computer produced by factory A is defective is 0.018, the probability that a computer produced by factory B is defective is 0.037, and the probability that a computer produced by factory C is defective is 0.036. A computer is selected at random and it is found to be defective. What is the probability it came from factory B? Answer: [ANS]", "answer_v3": [ "0.755102040816327" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0107", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability" ], "problem_v1": "Consider the experiment, called the birthday problem, where our task is to determine the probability that in a group of people of a certain size there are at least two people who have the same birthday (the same month and day of month). Suppose there is a room with 9 people in it, find the probability that at least two people have the same birthday. Ignore leap years; assume each year has 365 days. Answer=[ANS]", "answer_v1": [ "0.0946238" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the experiment, called the birthday problem, where our task is to determine the probability that in a group of people of a certain size there are at least two people who have the same birthday (the same month and day of month). Suppose there is a room with 5 people in it, find the probability that at least two people have the same birthday. Ignore leap years; assume each year has 365 days. Answer=[ANS]", "answer_v2": [ "0.0271356" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the experiment, called the birthday problem, where our task is to determine the probability that in a group of people of a certain size there are at least two people who have the same birthday (the same month and day of month). Suppose there is a room with 6 people in it, find the probability that at least two people have the same birthday. Ignore leap years; assume each year has 365 days. Answer=[ANS]", "answer_v3": [ "0.0404625" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0108", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "Bayes", "probability", "conditional" ], "problem_v1": "Scientific research on popular beverages consisted of 70 studies that were fully sponsored by the food industry, and 30 studies that were conducted with no corporate ties. Of those that were fully sponsored by the food industry, 13 \\% of the participants found the products unfavorable, 24 \\% were neutral, and 63 \\% found the products favorable. Of those that had no industry funding, 39 \\% found the products unfavorable, 14 \\% were neutral, and 47 \\% found the products favorable. What is the probability that a participant selected at random found the products favorable? [ANS]\nIf a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry? [ANS]\nIf a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding? [ANS]", "answer_v1": [ "0.582", "0.757731958762887", "0.5625" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Scientific research on popular beverages consisted of 55 studies that were fully sponsored by the food industry, and 45 studies that were conducted with no corporate ties. Of those that were fully sponsored by the food industry, 15 \\% of the participants found the products unfavorable, 21 \\% were neutral, and 64 \\% found the products favorable. Of those that had no industry funding, 37 \\% found the products unfavorable, 18 \\% were neutral, and 45 \\% found the products favorable. What is the probability that a participant selected at random found the products favorable? [ANS]\nIf a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry? [ANS]\nIf a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding? [ANS]", "answer_v2": [ "0.5545", "0.634806131650135", "0.668674698795181" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Scientific research on popular beverages consisted of 60 studies that were fully sponsored by the food industry, and 40 studies that were conducted with no corporate ties. Of those that were fully sponsored by the food industry, 13 \\% of the participants found the products unfavorable, 22 \\% were neutral, and 65 \\% found the products favorable. Of those that had no industry funding, 38 \\% found the products unfavorable, 14 \\% were neutral, and 48 \\% found the products favorable. What is the probability that a participant selected at random found the products favorable? [ANS]\nIf a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry? [ANS]\nIf a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding? [ANS]", "answer_v3": [ "0.582", "0.670103092783505", "0.660869565217391" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0109", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability" ], "problem_v1": "Real estate ads suggest that 58 \\% of homes for sale have garages, 33 \\% have swimming pools, and 22 \\% have both features. What is the probability that a home for sale has a) a pool or a garage? Answer=[ANS] \\% b) neither a pool nor a garage? Answer=[ANS] \\% c) a pool but no garage? Answer=[ANS] \\%", "answer_v1": [ "69", "31", "11" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Real estate ads suggest that 50 \\% of homes for sale have garages, 43 \\% have swimming pools, and 9 \\% have both features. What is the probability that a home for sale has a) a pool or a garage? Answer=[ANS] \\% b) neither a pool nor a garage? Answer=[ANS] \\% c) a pool but no garage? Answer=[ANS] \\%", "answer_v2": [ "84", "16", "34" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Real estate ads suggest that 53 \\% of homes for sale have garages, 33 \\% have swimming pools, and 12 \\% have both features. What is the probability that a home for sale has a) a pool or a garage? Answer=[ANS] \\% b) neither a pool nor a garage? Answer=[ANS] \\% c) a pool but no garage? Answer=[ANS] \\%", "answer_v3": [ "74", "26", "21" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0110", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability" ], "problem_v1": "(Note that an Ace is considered a face card for this problem) In drawing a single card from a regular deck of 52 cards we have:\n(a) P(Queen and a 3)=[ANS]\n(b) P(black or a 3)=[ANS]\n(c) P(black and a face card)=[ANS]\n(d) P(black or a face card)=[ANS]\n(e) P(black and a Queen)=[ANS]", "answer_v1": [ "0/52", "28/52", "8/52", "34/52", "2/52" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "(Note that an Ace is considered a face card for this problem) In drawing a single card from a regular deck of 52 cards we have:\n(a) P(black and a Queen)=[ANS]\n(b) P(black or a 3)=[ANS]\n(c) P(Queen and a 3)=[ANS]\n(d) P(face card or a number card)=[ANS]\n(e) P(black and a face card)=[ANS]", "answer_v2": [ "2/52", "28/52", "0/52", "52/52", "8/52" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "(Note that an Ace is considered a face card for this problem) In drawing a single card from a regular deck of 52 cards we have:\n(a) P(black and a face card)=[ANS]\n(b) P(Queen and a 3)=[ANS]\n(c) P(black or a face card)=[ANS]\n(d) P(black and a Queen)=[ANS]\n(e) P(face card or a number card)=[ANS]", "answer_v3": [ "8/52", "0/52", "34/52", "2/52", "52/52" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0111", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability", "conditional" ], "problem_v1": "An information technology company produces $48$ \\% of its computer chips at a plant in St. Louis and the remainder of its chips at a plant in Chicago. It is known that $0.85$ \\% of the chips produced in St. Louis are defective, while $1.05$ \\% of the chips produced at the plan in Chicago are defective. What is the probability that a randomly chosen computer chip produced by this company is defective and was produced in St. Louis? [ANS]", "answer_v1": [ "0.00408" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An information technology company produces $40$ \\% of its computer chips at a plant in St. Louis and the remainder of its chips at a plant in Chicago. It is known that $1.1$ \\% of the chips produced in St. Louis are defective, while $0.8$ \\% of the chips produced at the plan in Chicago are defective. What is the probability that a randomly chosen computer chip produced by this company is defective and was produced in St. Louis? [ANS]", "answer_v2": [ "0.0044" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An information technology company produces $43$ \\% of its computer chips at a plant in St. Louis and the remainder of its chips at a plant in Chicago. It is known that $0.85$ \\% of the chips produced in St. Louis are defective, while $0.9$ \\% of the chips produced at the plan in Chicago are defective. What is the probability that a randomly chosen computer chip produced by this company is defective and was produced in St. Louis? [ANS]", "answer_v3": [ "0.003655" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0112", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "probability", "conditional", "bayes" ], "problem_v1": "Suppose that $P(A)=0.48$, $P(C \\mid A)=0.01$, and $P(C' \\mid A')=0.0105$. Find $P(A \\mid C)$. [ANS]\n(Hint: Draw a tree diagram first)", "answer_v1": [ "0.0092425" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $P(A)=0.4$, $P(C \\mid A)=0.013$, and $P(C' \\mid A')=0.006$. Find $P(A \\mid C)$. [ANS]\n(Hint: Draw a tree diagram first)", "answer_v2": [ "0.00864362" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $P(A)=0.43$, $P(C \\mid A)=0.01$, and $P(C' \\mid A')=0.0075$. Find $P(A \\mid C)$. [ANS]\n(Hint: Draw a tree diagram first)", "answer_v3": [ "0.00754353" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0113", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "algebra", "probability", "conditional probability" ], "problem_v1": "A family has $6$ children. Assume that each child is as likely to be a boy as it is to be a girl. Find the probability that the family has $6$ girls if it is known the family has at least one girl. Answer: [ANS]", "answer_v1": [ "1/63" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A family has $3$ children. Assume that each child is as likely to be a boy as it is to be a girl. Find the probability that the family has $3$ girls if it is known the family has at least one girl. Answer: [ANS]", "answer_v2": [ "1/7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A family has $4$ children. Assume that each child is as likely to be a boy as it is to be a girl. Find the probability that the family has $4$ girls if it is known the family has at least one girl. Answer: [ANS]", "answer_v3": [ "1/15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0114", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "algebra", "probability" ], "problem_v1": "One hundred people were surveyed, and one question pertained to their educational background. The results of this question and their genders are given in the following table.\n$\\begin{array}{cccc}\\hline & Female (F) & Male (F\\,') & TotalTotal \\\\ \\hline College degree (D) & 30 & 20 & 50 \\\\ \\hline No college degree (D') & 30 & 20 & 50 \\\\ \\hline Total & 60 & 40 & 100 \\\\ \\hline \\end{array}$ If a person is selected at random from those surveyed, find the probability if each of the following events. 1. The person is female or has a college degree. Answer: [ANS]\n2. The person is male or does not have a college degree. Answer: [ANS]\n3. The person is female or does not have a college degree. Answer: [ANS]", "answer_v1": [ "1-20/100", "1-30/100", "1-20/100" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "One hundred people were surveyed, and one question pertained to their educational background. The results of this question and their genders are given in the following table.\n$\\begin{array}{cccc}\\hline & Female (F) & Male (F\\,') & TotalTotal \\\\ \\hline College degree (D) & 21 & 29 & 50 \\\\ \\hline No college degree (D') & 35 & 15 & 50 \\\\ \\hline Total & 56 & 44 & 100 \\\\ \\hline \\end{array}$ If a person is selected at random from those surveyed, find the probability if each of the following events. 1. The person is female or has a college degree. Answer: [ANS]\n2. The person is male or does not have a college degree. Answer: [ANS]\n3. The person is female or does not have a college degree. Answer: [ANS]", "answer_v2": [ "1-15/100", "1-21/100", "1-29/100" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "One hundred people were surveyed, and one question pertained to their educational background. The results of this question and their genders are given in the following table.\n$\\begin{array}{cccc}\\hline & Female (F) & Male (F\\,') & TotalTotal \\\\ \\hline College degree (D) & 24 & 26 & 50 \\\\ \\hline No college degree (D') & 30 & 20 & 50 \\\\ \\hline Total & 54 & 46 & 100 \\\\ \\hline \\end{array}$ If a person is selected at random from those surveyed, find the probability if each of the following events. 1. The person is female or has a college degree. Answer: [ANS]\n2. The person is male or does not have a college degree. Answer: [ANS]\n3. The person is female or does not have a college degree. Answer: [ANS]", "answer_v3": [ "1-20/100", "1-24/100", "1-26/100" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0115", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "algebra", "probability", "conditional probability" ], "problem_v1": "In a recent election there were $1000$ eligible voters. They were asked to vote on two issues, $A$ and $B.$ The results were as follows: $250$ people voted for $A,$ $500$ people voted for $B,$ and $50$ people voted for $A$ and $B.$ If one person is chosen at random from the $1000$ eligible voters, find the following probabilities: 1. The person voted for $A,$ given that he voted for $B.$ Answer: [ANS]\n2. The person voted for $B,$ given that he voted for $A.$ Answer: [ANS]", "answer_v1": [ "50/500", "50/250" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "In a recent election there were $1000$ eligible voters. They were asked to vote on two issues, $A$ and $B.$ The results were as follows: $100$ people voted for $A,$ $550$ people voted for $B,$ and $25$ people voted for $A$ and $B.$ If one person is chosen at random from the $1000$ eligible voters, find the following probabilities: 1. The person voted for $A,$ given that he voted for $B.$ Answer: [ANS]\n2. The person voted for $B,$ given that he voted for $A.$ Answer: [ANS]", "answer_v2": [ "25/550", "25/100" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "In a recent election there were $1000$ eligible voters. They were asked to vote on two issues, $A$ and $B.$ The results were as follows: $150$ people voted for $A,$ $500$ people voted for $B,$ and $25$ people voted for $A$ and $B.$ If one person is chosen at random from the $1000$ eligible voters, find the following probabilities: 1. The person voted for $A,$ given that he voted for $B.$ Answer: [ANS]\n2. The person voted for $B,$ given that he voted for $A.$ Answer: [ANS]", "answer_v3": [ "25/500", "25/150" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0116", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [], "problem_v1": "Angus goes to one of two coffee shops in his home town. He goes to Tarbucks 75 \\% of the time and otherwise goes to Costly Coffee. Either way, he buys a latte 67 \\% of the time, regardless of which place he chose.\nPart a) You are told that Angus went into town for a coffee today. What is the probability (to 3 decimal places) that he had a latte at Tarbucks? [ANS]\nPart b) Are the two events that gave the joint probability in (a) independent of each other? [ANS] A. Yes B. No\nPart c) Given that Angus had a latte in town, what is the probability (to 3 decimal places) that he drank at Costly Coffee? [ANS]\nPart d) What is the probability (to 3 decimal places) that Angus went to Tarbucks or had a latte or both? [ANS]", "answer_v1": [ "0.5025", "A", "0.25", "0.9175" ], "answer_type_v1": [ "NV", "MCS", "NV", "NV" ], "options_v1": [ [], [ "A", "B" ], [], [] ], "problem_v2": "Angus goes to one of two coffee shops in his home town. He goes to Tarbucks 61 \\% of the time and otherwise goes to Costly Coffee. Either way, he buys a latte 74 \\% of the time, regardless of which place he chose.\nPart a) You are told that Angus went into town for a coffee today. What is the probability (to 3 decimal places) that he had a latte at Tarbucks? [ANS]\nPart b) Are the two events that gave the joint probability in (a) independent of each other? [ANS] A. No B. Yes\nPart c) Given that Angus had a latte in town, what is the probability (to 3 decimal places) that he drank at Costly Coffee? [ANS]\nPart d) What is the probability (to 3 decimal places) that Angus went to Tarbucks or had a latte or both? [ANS]", "answer_v2": [ "0.4514", "B", "0.39", "0.8986" ], "answer_type_v2": [ "NV", "MCS", "NV", "NV" ], "options_v2": [ [], [ "A", "B" ], [], [] ], "problem_v3": "Angus goes to one of two coffee shops in his home town. He goes to Tarbucks 66 \\% of the time and otherwise goes to Costly Coffee. Either way, he buys a latte 67 \\% of the time, regardless of which place he chose.\nPart a) You are told that Angus went into town for a coffee today. What is the probability (to 3 decimal places) that he had a latte at Tarbucks? [ANS]\nPart b) Are the two events that gave the joint probability in (a) independent of each other? [ANS] A. No B. Yes\nPart c) Given that Angus had a latte in town, what is the probability (to 3 decimal places) that he drank at Costly Coffee? [ANS]\nPart d) What is the probability (to 3 decimal places) that Angus went to Tarbucks or had a latte or both? [ANS]", "answer_v3": [ "0.4422", "B", "0.34", "0.8878" ], "answer_type_v3": [ "NV", "MCS", "NV", "NV" ], "options_v3": [ [], [ "A", "B" ], [], [] ] }, { "id": "Probability_0117", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "Suppose that $A$ and $B$ are two events for which $P(A)=0.3$, $P(B)=0.72$, and $P(B|A)=0.47$ Find each of the following: A. $P(A\\;\\mathrm{and}\\;B)=$ [ANS]\nB. $P(A\\;\\mathrm{or}\\;B)=$ [ANS]\nC. $P(A|B)=$ [ANS]", "answer_v1": [ "0.141", "0.879", "0.195833333333333" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that $A$ and $B$ are two events for which $P(A)=0.16$, $P(B)=0.82$, and $P(B|A)=0.38$ Find each of the following: A. $P(A\\;\\mathrm{and}\\;B)=$ [ANS]\nB. $P(A\\;\\mathrm{or}\\;B)=$ [ANS]\nC. $P(A|B)=$ [ANS]", "answer_v2": [ "0.0608", "0.9192", "0.0741463414634146" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that $A$ and $B$ are two events for which $P(A)=0.21$, $P(B)=0.73$, and $P(B|A)=0.41$ Find each of the following: A. $P(A\\;\\mathrm{and}\\;B)=$ [ANS]\nB. $P(A\\;\\mathrm{or}\\;B)=$ [ANS]\nC. $P(A|B)=$ [ANS]", "answer_v3": [ "0.0861", "0.8539", "0.117945205479452" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0118", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "A firm has classified its customers in two ways: according to whether the account is overdue and whether the account is new (less than 12 months) or old. An analysis of the firm's records provided the input for the following table of joint probabilities:\n$\\begin{array}{ccc}\\hline & Overdue & Not overdue \\\\ \\hline New & 0.07 & 0.14 \\\\ \\hline Old & 0.52 & 0.27 \\\\ \\hline \\end{array}$\nOne account is selected at random. A. If the account is overdue, what is the probability that it is new? [ANS]\nB. If the account is new, what is the probability that it is overdue? [ANS]", "answer_v1": [ "0.11864406779661", "0.333333333333333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A firm has classified its customers in two ways: according to whether the account is overdue and whether the account is new (less than 12 months) or old. An analysis of the firm's records provided the input for the following table of joint probabilities:\n$\\begin{array}{ccc}\\hline & Overdue & Not overdue \\\\ \\hline New & 0.02 & 0.11 \\\\ \\hline Old & 0.55 & 0.32 \\\\ \\hline \\end{array}$\nOne account is selected at random. A. If the account is overdue, what is the probability that it is new? [ANS]\nB. If the account is new, what is the probability that it is overdue? [ANS]", "answer_v2": [ "0.0350877192982456", "0.153846153846154" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A firm has classified its customers in two ways: according to whether the account is overdue and whether the account is new (less than 12 months) or old. An analysis of the firm's records provided the input for the following table of joint probabilities:\n$\\begin{array}{ccc}\\hline & Overdue & Not overdue \\\\ \\hline New & 0.04 & 0.11 \\\\ \\hline Old & 0.52 & 0.33 \\\\ \\hline \\end{array}$\nOne account is selected at random. A. If the account is overdue, what is the probability that it is new? [ANS]\nB. If the account is new, what is the probability that it is overdue? [ANS]", "answer_v3": [ "0.0714285714285714", "0.266666666666667" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0119", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "An accounting professor claims that no more than one quarter of undergraduate business students will major in accounting. What is the probability that in a random sample of 750 undergraduate business students, 191 or more will major in accounting? Probability=[ANS]", "answer_v1": [ "0.3839412070415" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An accounting professor claims that no more than one quarter of undergraduate business students will major in accounting. What is the probability that in a random sample of 610 undergraduate business students, 171 or more will major in accounting? Probability=[ANS]", "answer_v2": [ "0.041829346561687" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An accounting professor claims that no more than one quarter of undergraduate business students will major in accounting. What is the probability that in a random sample of 660 undergraduate business students, 169 or more will major in accounting? Probability=[ANS]", "answer_v3": [ "0.359583140666589" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0120", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 53.5\\%. What is the probability that in a random sample of 520 voters, less than 49.3\\% say they will vote for the incumbent? Probability=[ANS]", "answer_v1": [ "0.0274160045752792" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 52.1\\%. What is the probability that in a random sample of 590 voters, less than 48.3\\% say they will vote for the incumbent? Probability=[ANS]", "answer_v2": [ "0.0323254854833332" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 52.6\\%. What is the probability that in a random sample of 520 voters, less than 48.5\\% say they will vote for the incumbent? Probability=[ANS]", "answer_v3": [ "0.0305741952602504" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0121", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "A manufacturer of aspirin claims that the proportion of headache sufferers who get relief with just two aspirins is 63.5\\%. What is the probability that in a random sample of 520 headache sufferers, less than 59.3\\% obtain relief? Probability=[ANS]", "answer_v1": [ "0.0233296570048441" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A manufacturer of aspirin claims that the proportion of headache sufferers who get relief with just two aspirins is 62.1\\%. What is the probability that in a random sample of 590 headache sufferers, less than 58.3\\% obtain relief? Probability=[ANS]", "answer_v2": [ "0.028547321114107" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A manufacturer of aspirin claims that the proportion of headache sufferers who get relief with just two aspirins is 62.6\\%. What is the probability that in a random sample of 520 headache sufferers, less than 58.5\\% obtain relief? Probability=[ANS]", "answer_v3": [ "0.0266645345136303" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0122", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "3", "keywords": [ "statistics", "probability" ], "problem_v1": "A psychologist believes that 80\\% of male drivers when lost continue to drive, hoping to find the location they seek rather than ask directions. To examine this belief, he took a random sample of 46 male drivers and asked each what they did when lost. If the belief is true, determine the probability that less than 72\\% said they continue driving. Probability=[ANS]", "answer_v1": [ "0.087475472637477" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A psychologist believes that 76\\% of male drivers when lost continue to drive, hoping to find the location they seek rather than ask directions. To examine this belief, he took a random sample of 50 male drivers and asked each what they did when lost. If the belief is true, determine the probability that less than 70\\% said they continue driving. Probability=[ANS]", "answer_v2": [ "0.160257708151404" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A psychologist believes that 77\\% of male drivers when lost continue to drive, hoping to find the location they seek rather than ask directions. To examine this belief, he took a random sample of 46 male drivers and asked each what they did when lost. If the belief is true, determine the probability that less than 71\\% said they continue driving. Probability=[ANS]", "answer_v3": [ "0.166775126069365" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0123", "subject": "Probability", "topic": "Sample Space", "subtopic": "Conditional probability -- direct", "level": "1", "keywords": [ "probability", "conditional" ], "problem_v1": "The chartered financial analyst (CFA) is a designation earned after taking three annual exams (CFA I,II, and III). The exams are taken in early June. Candidates who pass an exam are eligible to take the exam for the next level in the following year. The pass rates for levels I, II, and III are 0.57, 0.76, and 0.86, respectively. Suppose that 3,000 candidates take the level I exam, 2,500 take the level II exam and 2,000 take the level III exam. Suppose that one of the 7,500 candidates is selected at random. What is the probability that he or she passes the exam? Probability=[ANS]", "answer_v1": [ "0.710666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The chartered financial analyst (CFA) is a designation earned after taking three annual exams (CFA I,II, and III). The exams are taken in early June. Candidates who pass an exam are eligible to take the exam for the next level in the following year. The pass rates for levels I, II, and III are 0.5, 0.8, and 0.81, respectively. Suppose that 3,000 candidates take the level I exam, 2,500 take the level II exam and 2,000 take the level III exam. Suppose that one of the 7,500 candidates is selected at random. What is the probability that he or she passes the exam? Probability=[ANS]", "answer_v2": [ "0.682666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The chartered financial analyst (CFA) is a designation earned after taking three annual exams (CFA I,II, and III). The exams are taken in early June. Candidates who pass an exam are eligible to take the exam for the next level in the following year. The pass rates for levels I, II, and III are 0.53, 0.76, and 0.82, respectively. Suppose that 3,000 candidates take the level I exam, 2,500 take the level II exam and 2,000 take the level III exam. Suppose that one of the 7,500 candidates is selected at random. What is the probability that he or she passes the exam? Probability=[ANS]", "answer_v3": [ "0.684" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0124", "subject": "Probability", "topic": "Sample Space", "subtopic": "Independence", "level": "3", "keywords": [ "probability", "replacement" ], "problem_v1": "All that is left in a packet of candy are 8 reds, 5 greens, and 3 blues.\n(a)What is the probability that a random drawing yields a red followed by a green assuming that the first candy drawn is put back into the packet? Answer: [ANS]\n(b)Are the events 'red' and 'green' dependent? Answer: [ANS]", "answer_v1": [ "0.15625", "NO" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "All that is left in a packet of candy are 5 reds, 2 greens, and 4 blues.\n(a)What is the probability that a random drawing yields a red followed by a red assuming that the first candy drawn is put back into the packet? Answer: [ANS]\n(b)Are the events 'red' and 'red' independent? Answer: [ANS]", "answer_v2": [ "0.206611570247934", "YES" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "All that is left in a packet of candy are 6 reds, 3 greens, and 3 blues.\n(a)What is the probability that a random drawing yields a green followed by a blue assuming that the first candy drawn is put back into the packet? Answer: [ANS]\n(b)Are the events 'green' and 'blue' dependent? Answer: [ANS]", "answer_v3": [ "0.0625", "NO" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Probability_0125", "subject": "Probability", "topic": "Sample Space", "subtopic": "Independence", "level": "2", "keywords": [ "probability", "dependence" ], "problem_v1": "If A and B are mutually exclusive events with P(A)=0.70, then P(B): [ANS] A. cannot be larger than 0.30 B. cannot be smaller than 0.30 C. can be any value between 0 and 0.70 D. can be any value between 0 and 1\nA useful graphical method of constructing the sample space for an experiment is: [ANS] A. an ogive B. a pie chart C. a tree diagram D. a histogram", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "If A and B are mutually exclusive events with P(A)=0.70, then P(B): [ANS] A. can be any value between 0 and 0.70 B. cannot be smaller than 0.30 C. can be any value between 0 and 1 D. cannot be larger than 0.30\nIf P(A)=0.20, P(B)=0.30 and P(A and B)=0.06, then A and B are: [ANS] A. independent events B. complementary events C. dependent events D. mutually exclusive events", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "If A and B are mutually exclusive events with P(A)=0.70, then P(B): [ANS] A. can be any value between 0 and 0.70 B. cannot be larger than 0.30 C. cannot be smaller than 0.30 D. can be any value between 0 and 1\nTwo events A and B are said to be mutually exclusive if: [ANS] A. $P(A \\mbox{and} B)=0$ B. $P(B|A)=1$ C. $P(A|B)=1$ D. $P(A \\mbox{and} B)=1$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0126", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "2", "keywords": [ "Conditional", "Probability" ], "problem_v1": "Urn A has 8 white and 5 red balls. Urn B has 16 white and 12 red balls. We flip a fair coin. If the outcome is heads, then a ball from urn A is selected, whereas if the outcome is tails, then a ball from urn B is selected. Suppose that a white ball is selected. What is the probability that the coin landed heads? [ANS]", "answer_v1": [ "0.518518518518518" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Urn A has 15 white and 2 red balls. Urn B has 9 white and 10 red balls. We flip a fair coin. If the outcome is heads, then a ball from urn A is selected, whereas if the outcome is tails, then a ball from urn B is selected. Suppose that a red ball is selected. What is the probability that the coin landed heads? [ANS]", "answer_v2": [ "0.182692307692308" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Urn A has 3 white and 11 red balls. Urn B has 16 white and 8 red balls. We flip a fair coin. If the outcome is heads, then a ball from urn A is selected, whereas if the outcome is tails, then a ball from urn B is selected. Suppose that a red ball is selected. What is the probability that the coin landed heads? [ANS]", "answer_v3": [ "0.702127659574468" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0127", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "Bayes", "probability", "conditional" ], "problem_v1": "On average 68 \\% of Finite Mathematics students spend some time in the Mathematics Department's resource room. Half of these students spend more than 90 minutes per week in the resource room. At the end of the semester the students in the class were asked how many minutes per week they spent in the resource room and whether they passed or failed. The passing rates are summarized in the following table:\n$\\begin{array}{cc}\\hline Time spent in resource room & Pass \\% \\\\ \\hline None & 27 \\\\ \\hline Between 1 and 90 minutes & 48 \\\\ \\hline More than 90 minutes & 75 \\\\ \\hline \\end{array}$\nIf a randomly chosen student did not pass the course, what is the probability that he or she did not study in the resource room? Answer: [ANS]", "answer_v1": [ "0.4715381509891" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "On average 62 \\% of Finite Mathematics students spend some time in the Mathematics Department's resource room. Half of these students spend more than 90 minutes per week in the resource room. At the end of the semester the students in the class were asked how many minutes per week they spent in the resource room and whether they passed or failed. The passing rates are summarized in the following table:\n$\\begin{array}{cc}\\hline Time spent in resource room & Pass \\% \\\\ \\hline None & 31 \\\\ \\hline Between 1 and 90 minutes & 42 \\\\ \\hline More than 90 minutes & 69 \\\\ \\hline \\end{array}$\nIf a randomly chosen student did not pass the course, what is the probability that he or she did not study in the resource room? Answer: [ANS]", "answer_v2": [ "0.487270024159078" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "On average 64 \\% of Finite Mathematics students spend some time in the Mathematics Department's resource room. Half of these students spend more than 90 minutes per week in the resource room. At the end of the semester the students in the class were asked how many minutes per week they spent in the resource room and whether they passed or failed. The passing rates are summarized in the following table:\n$\\begin{array}{cc}\\hline Time spent in resource room & Pass \\% \\\\ \\hline None & 29 \\\\ \\hline Between 1 and 90 minutes & 46 \\\\ \\hline More than 90 minutes & 73 \\\\ \\hline \\end{array}$\nIf a randomly chosen student did not pass the course, what is the probability that he or she did not study in the resource room? Answer: [ANS]", "answer_v3": [ "0.496503496503496" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0128", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "probability", "conditional" ], "problem_v1": "It is estimated that approximately 8.4\\% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 96\\% of all adults over $40$ with diabetes as having the disease and incorrectly diagnoses 3\\% of all adults over $40$ without diabetes as having the disease. a) Find the probability that a randomly selected adult over $40$ does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called \"false positives\"). [ANS]\nb) Find the probability that a randomly selected adult of $40$ is diagnosed as not having diabetes. [ANS]\nc) Find the probability that a randomly selected adult over $40$ actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called \"false negatives\"). [ANS]\n(Note: it will be helpful to first draw an appropriate tree diagram modeling the situation)", "answer_v1": [ "0.02748", "0.89188", "0.00376732" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "It is estimated that approximately 8.13\\% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 98\\% of all adults over $40$ with diabetes as having the disease and incorrectly diagnoses 1.5\\% of all adults over $40$ without diabetes as having the disease. a) Find the probability that a randomly selected adult over $40$ does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called \"false positives\"). [ANS]\nb) Find the probability that a randomly selected adult of $40$ is diagnosed as not having diabetes. [ANS]\nc) Find the probability that a randomly selected adult over $40$ actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called \"false negatives\"). [ANS]\n(Note: it will be helpful to first draw an appropriate tree diagram modeling the situation)", "answer_v2": [ "0.0137805", "0.906546", "0.00179362" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "It is estimated that approximately 8.22\\% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 96\\% of all adults over $40$ with diabetes as having the disease and incorrectly diagnoses 2\\% of all adults over $40$ without diabetes as having the disease. a) Find the probability that a randomly selected adult over $40$ does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called \"false positives\"). [ANS]\nb) Find the probability that a randomly selected adult of $40$ is diagnosed as not having diabetes. [ANS]\nc) Find the probability that a randomly selected adult over $40$ actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called \"false negatives\"). [ANS]\n(Note: it will be helpful to first draw an appropriate tree diagram modeling the situation)", "answer_v3": [ "0.018356", "0.902732", "0.00364228" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0129", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "probability", "conditional", "bayes" ], "problem_v1": "In a random sample of 1,000 people, it is found that 8.3\\% have a liver ailment. Of those who have a liver ailment, 8\\% are heavy drinkers, 65\\% are moderate drinkers, and 27\\% are nondrinkers. Of those who do not have a liver ailment, 13\\% are heavy drinkers, 43\\% are moderate drinkers, and 44\\% are nondrinkers. If a person is chosen at random, and he or she is a heavy drinker, what is the empirical probability of that person having a liver ailment? [ANS]\n(Hint: Draw a tree diagram first)", "answer_v1": [ "0.0527612" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In a random sample of 1,000 people, it is found that 6.2\\% have a liver ailment. Of those who have a liver ailment, 10\\% are heavy drinkers, 50\\% are moderate drinkers, and 40\\% are nondrinkers. Of those who do not have a liver ailment, 10\\% are heavy drinkers, 50\\% are moderate drinkers, and 40\\% are nondrinkers. If a person is chosen at random, and he or she is a heavy drinker, what is the empirical probability of that person having a liver ailment? [ANS]\n(Hint: Draw a tree diagram first)", "answer_v2": [ "0.062" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In a random sample of 1,000 people, it is found that 6.9\\% have a liver ailment. Of those who have a liver ailment, 9\\% are heavy drinkers, 55\\% are moderate drinkers, and 36\\% are nondrinkers. Of those who do not have a liver ailment, 12\\% are heavy drinkers, 42\\% are moderate drinkers, and 46\\% are nondrinkers. If a person is chosen at random, and he or she is a heavy drinker, what is the empirical probability of that person having a liver ailment? [ANS]\n(Hint: Draw a tree diagram first)", "answer_v3": [ "0.0526584" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0130", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "probability", "conditional", "bayes" ], "problem_v1": "A biomedical research company produces $48 \\%$ of its insulin at a plant in Kansas City, and the remainder is produced at a plant in Jefferson City. Quality control has shown that $1 \\%$ of the insulin produced at the plant in Kansas City is defective, while $1.05 \\%$ of the insulin produced at the plant in Jefferson City is defective. What is the probability that a randomly chosen unit of insulin came from the plant in Jefferson City given that it is defective? [ANS]\n(Hint: Draw a tree diagram first)", "answer_v1": [ "0.532164" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A biomedical research company produces $40 \\%$ of its insulin at a plant in Kansas City, and the remainder is produced at a plant in Jefferson City. Quality control has shown that $1.3 \\%$ of the insulin produced at the plant in Kansas City is defective, while $0.6 \\%$ of the insulin produced at the plant in Jefferson City is defective. What is the probability that a randomly chosen unit of insulin came from the plant in Jefferson City given that it is defective? [ANS]\n(Hint: Draw a tree diagram first)", "answer_v2": [ "0.409091" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A biomedical research company produces $43 \\%$ of its insulin at a plant in Kansas City, and the remainder is produced at a plant in Jefferson City. Quality control has shown that $1 \\%$ of the insulin produced at the plant in Kansas City is defective, while $0.75 \\%$ of the insulin produced at the plant in Jefferson City is defective. What is the probability that a randomly chosen unit of insulin came from the plant in Jefferson City given that it is defective? [ANS]\n(Hint: Draw a tree diagram first)", "answer_v3": [ "0.498542" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0131", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [], "problem_v1": "A breathalyser test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. The device is not totally reliable: 7 \\% of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit, while 10 \\% of drivers who are above the legal limit will give a reading below that level. Suppose that in fact 15 \\% of drivers are above the legal alcohol limit, and the police stop a driver at random. Give answers to the following to four decimal places.\nPart a) What is the probability that the driver is incorrectly classified as being over the limit? [ANS]\nPart b) What is the probability that the driver is correctly classified as being over the limit? [ANS]\nPart c) Find the probability that the driver gives a breathalyser test reading that is over the limit. [ANS]\nPart d) Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit. [ANS]", "answer_v1": [ "0.0595", "0.135", "0.1945", "0.9814" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A breathalyser test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. The device is not totally reliable: 3 \\% of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit, while 10 \\% of drivers who are above the legal limit will give a reading below that level. Suppose that in fact 18 \\% of drivers are above the legal alcohol limit, and the police stop a driver at random. Give answers to the following to four decimal places.\nPart a) What is the probability that the driver is incorrectly classified as being over the limit? [ANS]\nPart b) What is the probability that the driver is correctly classified as being over the limit? [ANS]\nPart c) Find the probability that the driver gives a breathalyser test reading that is over the limit. [ANS]\nPart d) Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit. [ANS]", "answer_v2": [ "0.0246", "0.162", "0.1866", "0.9779" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A breathalyser test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. The device is not totally reliable: 4 \\% of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit, while 10 \\% of drivers who are above the legal limit will give a reading below that level. Suppose that in fact 15 \\% of drivers are above the legal alcohol limit, and the police stop a driver at random. Give answers to the following to four decimal places.\nPart a) What is the probability that the driver is incorrectly classified as being over the limit? [ANS]\nPart b) What is the probability that the driver is correctly classified as being over the limit? [ANS]\nPart c) Find the probability that the driver gives a breathalyser test reading that is over the limit. [ANS]\nPart d) Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit. [ANS]", "answer_v3": [ "0.034", "0.135", "0.169", "0.9819" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0132", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "probability", "conditional" ], "problem_v1": "Suppose that at UVA, 72\\% of all undergraduates are in the College, 12\\% are in Engineering, 7\\% are in Commerce, 4\\% are in Nursing, and 5\\% are in Architecture. In each school, the percentage of females is as follows: 56\\% in the College, 24\\% in Engineering, 46\\% in Commerce, 88\\% in Nursing, and 33\\% in Architecture. If a randomly selected student is male, what is the probability that he's from the College? Probability=[ANS]", "answer_v1": [ "0.65441024581698" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that at UVA, 76\\% of all undergraduates are in the College, 9\\% are in Engineering, 8\\% are in Commerce, 3\\% are in Nursing, and 4\\% are in Architecture. In each school, the percentage of females is as follows: 57\\% in the College, 27\\% in Engineering, 46\\% in Commerce, 86\\% in Nursing, and 31\\% in Architecture. If a randomly selected student is male, what is the probability that he's from the College? Probability=[ANS]", "answer_v2": [ "0.69903743315508" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that at UVA, 76\\% of all undergraduates are in the College, 10\\% are in Engineering, 7\\% are in Commerce, 3\\% are in Nursing, and 4\\% are in Architecture. In each school, the percentage of females is as follows: 59\\% in the College, 24\\% in Engineering, 46\\% in Commerce, 89\\% in Nursing, and 35\\% in Architecture. If a randomly selected student is male, what is the probability that he's from the College? Probability=[ANS]", "answer_v3": [ "0.685287002419177" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0133", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "probability", "conditional" ], "problem_v1": "Bad gums may mean a bad heart. Researchers discovered that 83\\% of people who have suffered a heart attack had periodontal disease, an inflammation of the gums. Only 31\\% of healthy people have this disease. Suppose that in a certain community heart attacks are quite rare, occurring with only 13\\% probability. A. If someone has periodontal disease, what is the probability that he or she will have a heart attack? Probability=[ANS]\nB. If 42\\% of the people in a community will have a heart attack, what is the probability that a person with periodontal disease will have a heart attack? Probability=[ANS]", "answer_v1": [ "0.285752118644068", "0.659727479182438" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Bad gums may mean a bad heart. Researchers discovered that 75\\% of people who have suffered a heart attack had periodontal disease, an inflammation of the gums. Only 35\\% of healthy people have this disease. Suppose that in a certain community heart attacks are quite rare, occurring with only 10\\% probability. A. If someone has periodontal disease, what is the probability that he or she will have a heart attack? Probability=[ANS]\nB. If 38\\% of the people in a community will have a heart attack, what is the probability that a person with periodontal disease will have a heart attack? Probability=[ANS]", "answer_v2": [ "0.192307692307692", "0.567729083665339" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Bad gums may mean a bad heart. Researchers discovered that 78\\% of people who have suffered a heart attack had periodontal disease, an inflammation of the gums. Only 31\\% of healthy people have this disease. Suppose that in a certain community heart attacks are quite rare, occurring with only 11\\% probability. A. If someone has periodontal disease, what is the probability that he or she will have a heart attack? Probability=[ANS]\nB. If 41\\% of the people in a community will have a heart attack, what is the probability that a person with periodontal disease will have a heart attack? Probability=[ANS]", "answer_v3": [ "0.237213160077412", "0.63616471056296" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0134", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "probability", "conditional" ], "problem_v1": "Three airlines serve a small town in Ohio. Airline A has 53\\% of all scheduled flights, airline B has 31\\% and airline C has the remaining 16\\%. Their on-time rates are 81\\%, 67\\%, and 38\\%, respectively. A flight just left on-time. What is the probability that it was a flight of airline A? Probability=[ANS]", "answer_v1": [ "0.615219260533104" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Three airlines serve a small town in Ohio. Airline A has 45\\% of all scheduled flights, airline B has 35\\% and airline C has the remaining 20\\%. Their on-time rates are 76\\%, 63\\%, and 45\\%, respectively. A flight just left on-time. What is the probability that it was a flight of airline A? Probability=[ANS]", "answer_v2": [ "0.524137931034483" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Three airlines serve a small town in Ohio. Airline A has 48\\% of all scheduled flights, airline B has 31\\% and airline C has the remaining 21\\%. Their on-time rates are 78\\%, 66\\%, and 37\\%, respectively. A flight just left on-time. What is the probability that it was a flight of airline A? Probability=[ANS]", "answer_v3": [ "0.570123343992691" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0135", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "4", "keywords": [ "probability", "conditional" ], "problem_v1": "Data from Office on Smoking and Health, Centers for Disease Control and Prevention, indicate that 43\\% of adults who did not finish high school, 33\\% of high school graduates, 25\\% of adults who completed some college, and 14\\% of college graduates smoke. Suppose that one individual is selected at random and it is discovered that the individual smokes. Use the probabilities in the following table to calculate the probability that the individual is a college graduate.\n$\\begin{array}{ccc}\\hline Education & Employed & Unemployed \\\\ \\hline Not a high school graduate & 0.0975 & 0.0080 \\\\ \\hline High school graduate & 0.3108 & 0.0128 \\\\ \\hline Some college, no degree & 0.1785 & 0.0062 \\\\ \\hline Associate Degree & 0.0849 & 0.0023 \\\\ \\hline Bachelor Degree & 0.1959 & 0.0041 \\\\ \\hline Advanced Degree & 0.0975 & 0.0015 \\\\ \\hline \\end{array}$\nProbability=[ANS]\nHints: This problem has all the information you need, but not in the typical ready-to-use form. The table above can tell you the proportion of people with various levels of education in the population. Keep in mind that any degree (Associate, Bachelor, or Advanced) counts as graduating from college.", "answer_v1": [ "0.214218925814989" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Data from Office on Smoking and Health, Centers for Disease Control and Prevention, indicate that 35\\% of adults who did not finish high school, 35\\% of high school graduates, 22\\% of adults who completed some college, and 12\\% of college graduates smoke. Suppose that one individual is selected at random and it is discovered that the individual smokes. Use the probabilities in the following table to calculate the probability that the individual is a college graduate.\n$\\begin{array}{ccc}\\hline Education & Employed & Unemployed \\\\ \\hline Not a high school graduate & 0.0975 & 0.0080 \\\\ \\hline High school graduate & 0.3108 & 0.0128 \\\\ \\hline Some college, no degree & 0.1785 & 0.0062 \\\\ \\hline Associate Degree & 0.0849 & 0.0023 \\\\ \\hline Bachelor Degree & 0.1959 & 0.0041 \\\\ \\hline Advanced Degree & 0.0975 & 0.0015 \\\\ \\hline \\end{array}$\nProbability=[ANS]\nHints: This problem has all the information you need, but not in the typical ready-to-use form. The table above can tell you the proportion of people with various levels of education in the population. Keep in mind that any degree (Associate, Bachelor, or Advanced) counts as graduating from college.", "answer_v2": [ "0.195409907953602" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Data from Office on Smoking and Health, Centers for Disease Control and Prevention, indicate that 38\\% of adults who did not finish high school, 33\\% of high school graduates, 23\\% of adults who completed some college, and 13\\% of college graduates smoke. Suppose that one individual is selected at random and it is discovered that the individual smokes. Use the probabilities in the following table to calculate the probability that the individual is a college graduate.\n$\\begin{array}{ccc}\\hline Education & Employed & Unemployed \\\\ \\hline Not a high school graduate & 0.0975 & 0.0080 \\\\ \\hline High school graduate & 0.3108 & 0.0128 \\\\ \\hline Some college, no degree & 0.1785 & 0.0062 \\\\ \\hline Associate Degree & 0.0849 & 0.0023 \\\\ \\hline Bachelor Degree & 0.1959 & 0.0041 \\\\ \\hline Advanced Degree & 0.0975 & 0.0015 \\\\ \\hline \\end{array}$\nProbability=[ANS]\nHints: This problem has all the information you need, but not in the typical ready-to-use form. The table above can tell you the proportion of people with various levels of education in the population. Keep in mind that any degree (Associate, Bachelor, or Advanced) counts as graduating from college.", "answer_v3": [ "0.209571515037672" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0136", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "3", "keywords": [ "probability", "conditional" ], "problem_v1": "Your favorite team is in the World Series. You have assigned a probability of 63\\% that they will win the championship. Past records indicate that when teams win the championship, they win the first game of the series 71\\% of the time. When they lose the championship, they win the first game 26\\% of the time. The first game is over and your team has lost. What is the probability that they will win the World Series? Probability=[ANS]", "answer_v1": [ "0.400219058050383" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Your favorite team is in the World Series. You have assigned a probability of 55\\% that they will win the championship. Past records indicate that when teams win the championship, they win the first game of the series 75\\% of the time. When they lose the championship, they win the first game 21\\% of the time. The first game is over and your team has lost. What is the probability that they will win the World Series? Probability=[ANS]", "answer_v2": [ "0.278904665314402" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Your favorite team is in the World Series. You have assigned a probability of 58\\% that they will win the championship. Past records indicate that when teams win the championship, they win the first game of the series 71\\% of the time. When they lose the championship, they win the first game 23\\% of the time. The first game is over and your team has lost. What is the probability that they will win the World Series? Probability=[ANS]", "answer_v3": [ "0.342148087876322" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0137", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "4", "keywords": [ "probability", "conditional" ], "problem_v1": "Transplant operations have become routine and one common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ. There are several new drugs available for such circumstances and the earlier the drug is administered, the higher the probability of averting rejection. The New England Journal of Medicine recently reported the development of a new urine test to detect early warning signs that the body is rejecting a transplanted kidney. However, like most other tests, the new test is not perfect. In fact, 20\\% of people who do reject the transplant test negative, and 9\\% of people who do not reject the transplant test positive. Physicians know that in about 31\\% of kidney transplants the body tries to reject the organ. If the new test has a positive result (indicating early warning of rejection), what is the probability that the body is attempting to reject the kidney? Probability=[ANS]", "answer_v1": [ "0.799742018703644" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Transplant operations have become routine and one common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ. There are several new drugs available for such circumstances and the earlier the drug is administered, the higher the probability of averting rejection. The New England Journal of Medicine recently reported the development of a new urine test to detect early warning signs that the body is rejecting a transplanted kidney. However, like most other tests, the new test is not perfect. In fact, 20\\% of people who do reject the transplant test negative, and 5\\% of people who do not reject the transplant test positive. Physicians know that in about 35\\% of kidney transplants the body tries to reject the organ. If the new test has a positive result (indicating early warning of rejection), what is the probability that the body is attempting to reject the kidney? Probability=[ANS]", "answer_v2": [ "0.896" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Transplant operations have become routine and one common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ. There are several new drugs available for such circumstances and the earlier the drug is administered, the higher the probability of averting rejection. The New England Journal of Medicine recently reported the development of a new urine test to detect early warning signs that the body is rejecting a transplanted kidney. However, like most other tests, the new test is not perfect. In fact, 20\\% of people who do reject the transplant test negative, and 6\\% of people who do not reject the transplant test positive. Physicians know that in about 31\\% of kidney transplants the body tries to reject the organ. If the new test has a positive result (indicating early warning of rejection), what is the probability that the body is attempting to reject the kidney? Probability=[ANS]", "answer_v3": [ "0.856945404284727" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0138", "subject": "Probability", "topic": "Sample Space", "subtopic": "Bayes theorem -- inverse probability", "level": "4", "keywords": [ "probability", "conditional" ], "problem_v1": "The U.S. National Highway Traffic Safety Administration gathers data concerning the causes of highway crashes where at least one fatality has occurred. From the 1998 annual study, the following probabilities were determined (BAC is blood-alcohol level): $P(BAC=0 | \\mbox{Crash with fatality})=0.638$ $P(BAC \\mbox{is between}.01 \\mbox{and}.09 | \\mbox{Crash with fatality})=0.329$ $P(BAC \\mbox{is greater than}.09 | \\mbox{Crash with fatality})=0.081$ Suppose over a certain stretch of highway during a 1-year period, the probability of being involved in a crash that results in at least one fatality is 0.024. It has been estimated that 11\\% of all drivers drive while their BAC is grater than.09. Determine the probability of a crash with at least one fatality if a driver drives while legally intoxicated (BAC greater than 0.09). Probability=[ANS]", "answer_v1": [ "0.0176727272727273" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The U.S. National Highway Traffic Safety Administration gathers data concerning the causes of highway crashes where at least one fatality has occurred. From the 1998 annual study, the following probabilities were determined (BAC is blood-alcohol level): $P(BAC=0 | \\mbox{Crash with fatality})=0.604$ $P(BAC \\mbox{is between}.01 \\mbox{and}.09 | \\mbox{Crash with fatality})=0.346$ $P(BAC \\mbox{is greater than}.09 | \\mbox{Crash with fatality})=0.057$ Suppose over a certain stretch of highway during a 1-year period, the probability of being involved in a crash that results in at least one fatality is 0.016. It has been estimated that 15\\% of all drivers drive while their BAC is grater than.09. Determine the probability of a crash with at least one fatality if a driver drives while legally intoxicated (BAC greater than 0.09). Probability=[ANS]", "answer_v2": [ "0.00608" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The U.S. National Highway Traffic Safety Administration gathers data concerning the causes of highway crashes where at least one fatality has occurred. From the 1998 annual study, the following probabilities were determined (BAC is blood-alcohol level): $P(BAC=0 | \\mbox{Crash with fatality})=0.615$ $P(BAC \\mbox{is between}.01 \\mbox{and}.09 | \\mbox{Crash with fatality})=0.33$ $P(BAC \\mbox{is greater than}.09 | \\mbox{Crash with fatality})=0.064$ Suppose over a certain stretch of highway during a 1-year period, the probability of being involved in a crash that results in at least one fatality is 0.02. It has been estimated that 11\\% of all drivers drive while their BAC is grater than.09. Determine the probability of a crash with at least one fatality if a driver drives while legally intoxicated (BAC greater than 0.09). Probability=[ANS]", "answer_v3": [ "0.0116363636363636" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0139", "subject": "Probability", "topic": "Sample Space", "subtopic": "Odds", "level": "2", "keywords": [ "Probability", "Events" ], "problem_v1": "If the probability of rain is $\\frac{7}{11}$, what are the odds against rain?\nAnswer: [ANS] to [ANS]", "answer_v1": [ "4", "7" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If the probability of rain is $\\frac{1}{5}$, what are the odds against rain?\nAnswer: [ANS] to [ANS]", "answer_v2": [ "4", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If the probability of rain is $\\frac{3}{7}$, what are the odds against rain?\nAnswer: [ANS] to [ANS]", "answer_v3": [ "4", "3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0140", "subject": "Probability", "topic": "Sample Space", "subtopic": "Odds", "level": "2", "keywords": [ "Probability", "Events" ], "problem_v1": "(a) If the odds for A are 5 to 6, what is the probability of A? Answer: [ANS]\n(b) If the odds against B are 5 do 4 what is the probability of B? Answer: [ANS]", "answer_v1": [ "0.454545454545455", "0.444444444444444" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) If the odds for A are 1 to 8, what is the probability of A? Answer: [ANS]\n(b) If the odds against B are 2 do 2 what is the probability of B? Answer: [ANS]", "answer_v2": [ "0.111111111111111", "0.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) If the odds for A are 2 to 6, what is the probability of A? Answer: [ANS]\n(b) If the odds against B are 2 do 3 what is the probability of B? Answer: [ANS]", "answer_v3": [ "0.25", "0.6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0141", "subject": "Probability", "topic": "Sample Space", "subtopic": "Odds", "level": "2", "keywords": [ "Probability", "Events" ], "problem_v1": "If $P(E)=\\frac{1}{8},$ find:\na) The odds FOR E. Answer: [ANS] to [ANS]\nb) The odds AGAINST E. Answer: [ANS] to [ANS]", "answer_v1": [ "1", "7", "7", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "If $P(E)=\\frac{1}{3},$ find:\na) The odds FOR E. Answer: [ANS] to [ANS]\nb) The odds AGAINST E. Answer: [ANS] to [ANS]", "answer_v2": [ "1", "2", "2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "If $P(E)=\\frac{1}{5},$ find:\na) The odds FOR E. Answer: [ANS] to [ANS]\nb) The odds AGAINST E. Answer: [ANS] to [ANS]", "answer_v3": [ "1", "4", "4", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0142", "subject": "Probability", "topic": "Sample Space", "subtopic": "Odds", "level": "2", "keywords": [ "probability", "union", "intersection", "complement" ], "problem_v1": "Given that the odds again an event $E$ are $23: 32$, find $P(E)$. [ANS]", "answer_v1": [ "0.581818" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given that the odds again an event $E$ are $3: 23$, find $P(E)$. [ANS]", "answer_v2": [ "0.884615" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given that the odds again an event $E$ are $10: 31$, find $P(E)$. [ANS]", "answer_v3": [ "0.756098" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0143", "subject": "Probability", "topic": "Sample Space", "subtopic": "Odds", "level": "2", "keywords": [ "probability", "union", "intersection", "complement" ], "problem_v1": "Find the odds in favor of selecting an ace or a diamond when one card is selected. [ANS] $:$ [ANS]", "answer_v1": [ "4", "9" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the odds in favor of rolling a total of seven when two dice are tossed. [ANS] $:$ [ANS]", "answer_v2": [ "1", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the odds against rolling an $8$ or higher when two dice are tossed. [ANS] $:$ [ANS]", "answer_v3": [ "21", "15" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0144", "subject": "Probability", "topic": "Sample Space", "subtopic": "Odds", "level": "3", "keywords": [], "problem_v1": "The presence of colorectal cancer can be tested via a fecal occult blood count test. Sometimes the test can give a positive result when the cancer is absent and sometimes the test records a negative when in fact the cancer is present. In a population of 10,000 adults screened for colorectal cancer via a fecal blood test, the following counts were recorded. $\\begin{array}{ccc}\\hline & Cancer Present & No Cancer \\\\ \\hline Test positive & 19 & 295 \\\\ \\hline Test negative & 11 & 9675 \\\\ \\hline \\end{array}$ Based on this information, find (to 3 decimal places) the odds that a person in the population has colorectal cancer given that their fecal occult blood test was positive. [ANS]", "answer_v1": [ "0.064" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The presence of colorectal cancer can be tested via a fecal occult blood count test. Sometimes the test can give a positive result when the cancer is absent and sometimes the test records a negative when in fact the cancer is present. In a population of 10,000 adults screened for colorectal cancer via a fecal blood test, the following counts were recorded. $\\begin{array}{ccc}\\hline & Cancer Present & No Cancer \\\\ \\hline Test positive & 15 & 299 \\\\ \\hline Test negative & 15 & 9671 \\\\ \\hline \\end{array}$ Based on this information, find (to 3 decimal places) the odds that a person in the population has colorectal cancer given that their fecal occult blood test was positive. [ANS]", "answer_v2": [ "0.05" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The presence of colorectal cancer can be tested via a fecal occult blood count test. Sometimes the test can give a positive result when the cancer is absent and sometimes the test records a negative when in fact the cancer is present. In a population of 10,000 adults screened for colorectal cancer via a fecal blood test, the following counts were recorded. $\\begin{array}{ccc}\\hline & Cancer Present & No Cancer \\\\ \\hline Test positive & 16 & 298 \\\\ \\hline Test negative & 14 & 9672 \\\\ \\hline \\end{array}$ Based on this information, find (to 3 decimal places) the odds that a person in the population has colorectal cancer given that their fecal occult blood test was positive. [ANS]", "answer_v3": [ "0.054" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0145", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "financial mathematics", "expect value" ], "problem_v1": "A game of chance involves rolling an unevenly balanced 4-sided die. The probability that a roll comes up 1 is 0.21, the probability that a roll comes up 1 or 2 is 0.47, and the probability that a roll comes up 2 or 3 is 0.51. If you win the amount that appears on the die, what is your expected winnings? (Note that the die has 4 sides.)\nAnswer=[ANS] dollars.", "answer_v1": [ "2.6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A game of chance involves rolling an unevenly balanced 4-sided die. The probability that a roll comes up 1 is 0.12, the probability that a roll comes up 1 or 2 is 0.51, and the probability that a roll comes up 2 or 3 is 0.46. If you win the amount that appears on the die, what is your expected winnings? (Note that the die has 4 sides.)\nAnswer=[ANS] dollars.", "answer_v2": [ "2.79" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A game of chance involves rolling an unevenly balanced 4-sided die. The probability that a roll comes up 1 is 0.15, the probability that a roll comes up 1 or 2 is 0.47, and the probability that a roll comes up 2 or 3 is 0.48. If you win the amount that appears on the die, what is your expected winnings? (Note that the die has 4 sides.)\nAnswer=[ANS] dollars.", "answer_v3": [ "2.75" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0146", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "Random Variable", "Expected Value" ], "problem_v1": "Four buses carrying $155$ high school students arrive to Montreal. The buses carry, respectively, $38$, $46$, $31$, and $40$ students. One of the studetns is randomly selected. Let $X$ denote the number of students that were on the bus carrying this randomly selected student. One of the $4$ bus drivers is also randomly selected. Let $Y$ denote the number of students on his bus. Compute the expectations of $X$ and $Y$: $E(X)=$ [ANS]\n$E(Y)=$ [ANS]", "answer_v1": [ "39.4903225806452", "38.75" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Four buses carrying $147$ high school students arrive to Montreal. The buses carry, respectively, $30$, $50$, $26$, and $41$ students. One of the studetns is randomly selected. Let $X$ denote the number of students that were on the bus carrying this randomly selected student. One of the $4$ bus drivers is also randomly selected. Let $Y$ denote the number of students on his bus. Compute the expectations of $X$ and $Y$: $E(X)=$ [ANS]\n$E(Y)=$ [ANS]", "answer_v2": [ "39.1632653061224", "36.75" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Four buses carrying $152$ high school students arrive to Montreal. The buses carry, respectively, $33$, $47$, $28$, and $44$ students. One of the studetns is randomly selected. Let $X$ denote the number of students that were on the bus carrying this randomly selected student. One of the $4$ bus drivers is also randomly selected. Let $Y$ denote the number of students on his bus. Compute the expectations of $X$ and $Y$: $E(X)=$ [ANS]\n$E(Y)=$ [ANS]", "answer_v3": [ "39.5921052631579", "38" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0147", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "Random Variable" ], "problem_v1": "A rock concert producer has scheduled an outdoor concert. The producer estimates the attendance will depend on the weather according to the following table. $\\begin{array}{ccc}\\hline Weather & Attendance & Probability \\\\ \\hline wet, cold & 6000 & 0.2 \\\\ \\hline wet, warm & 25000 & 0.2 \\\\ \\hline dry, cold & 20000 & 0.1 \\\\ \\hline dry, warm & 45000 & 0.5 \\\\ \\hline \\end{array}$\n(a) What is the expected attendance? answer: [ANS]\n(b) If tickets cost \\$ 20 each, the band will cost \\$ 250000, plus \\$ 45000 for administration. What is the expected profit? answer: [ANS]", "answer_v1": [ "30700", "319000" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A rock concert producer has scheduled an outdoor concert. The producer estimates the attendance will depend on the weather according to the following table. $\\begin{array}{ccc}\\hline Weather & Attendance & Probability \\\\ \\hline wet, cold & 3000 & 0.2 \\\\ \\hline wet, warm & 30000 & 0.1 \\\\ \\hline dry, cold & 15000 & 0.1 \\\\ \\hline dry, warm & 45000 & 0.6 \\\\ \\hline \\end{array}$\n(a) What is the expected attendance? answer: [ANS]\n(b) If tickets cost \\$ 15 each, the band will cost \\$ 250000, plus \\$ 40000 for administration. What is the expected profit? answer: [ANS]", "answer_v2": [ "32100", "191500" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A rock concert producer has scheduled an outdoor concert. The producer estimates the attendance will depend on the weather according to the following table. $\\begin{array}{ccc}\\hline Weather & Attendance & Probability \\\\ \\hline wet, cold & 4000 & 0.1 \\\\ \\hline wet, warm & 25000 & 0.1 \\\\ \\hline dry, cold & 20000 & 0.2 \\\\ \\hline dry, warm & 50000 & 0.6 \\\\ \\hline \\end{array}$\n(a) What is the expected attendance? answer: [ANS]\n(b) If tickets cost \\$ 30 each, the band will cost \\$ 300000, plus \\$ 45000 for administration. What is the expected profit? answer: [ANS]", "answer_v3": [ "36900", "762000" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0148", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "Expected Value" ], "problem_v1": "A fair die is rolled $16$ times. What is the expected sum of the $16$ rolls? [ANS]", "answer_v1": [ "56" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A fair die is rolled $4$ times. What is the expected sum of the $4$ rolls? [ANS]", "answer_v2": [ "14" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A fair die is rolled $8$ times. What is the expected sum of the $8$ rolls? [ANS]", "answer_v3": [ "28" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0149", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "probability", "expected value", "probability model" ], "problem_v1": "Below is a probability model.\n$\\begin{array}{ccccc}\\hline Outcome & 3.5 & 4 & 4.5 & 4.5 \\\\ \\hline Probability & 0.15 & 0.15 & 0.15 & 0.55 \\\\ \\hline \\end{array}$\nFind the expected value of the probability model. [ANS]", "answer_v1": [ "4.275" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Below is a probability model.\n$\\begin{array}{ccccc}\\hline Outcome & 0.5 & 0.5 & 2 & 6 \\\\ \\hline Probability & 0.4 & 0.15 & 0.15 & 0.3 \\\\ \\hline \\end{array}$\nFind the expected value of the probability model. [ANS]", "answer_v2": [ "2.375" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Below is a probability model.\n$\\begin{array}{ccccc}\\hline Outcome & 1.5 & 2 & 3.5 & 3.5 \\\\ \\hline Probability & 0.1 & 0.15 & 0.15 & 0.6 \\\\ \\hline \\end{array}$\nFind the expected value of the probability model. [ANS]", "answer_v3": [ "3.075" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0150", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "probability" ], "problem_v1": "Mark draws one card from a standard deck of 52. He receives \\$ 0.45 for a heart, \\$ 0.65 for a jack and \\$ 0.90 for the jack of hearts. How much should he pay for one draw? Answer: \\$ [ANS]", "answer_v1": [ "0.158653846153846" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Mark draws one card from a standard deck of 52. He receives \\$ 0.30 for a diamond, \\$ 0.70 for a king and \\$ 0.75 for the king of diamonds. How much should he pay for one draw? Answer: \\$ [ANS]", "answer_v2": [ "0.124038461538462" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Mark draws one card from a standard deck of 52. He receives \\$ 0.35 for a heart, \\$ 0.65 for an ace and \\$ 0.80 for the ace of hearts. How much should he pay for one draw? Answer: \\$ [ANS]", "answer_v3": [ "0.133653846153846" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0151", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "probability" ], "problem_v1": "Nick's Ski Rental rents skis, boots, and poles for \\$ 24 per day. The daily cost per set of skiis is \\$ 8. It includes maintenance, storage, and overhead. Daily profits depend on daily demand for skis and the number of sets available. Nick knows that on a typical weekend the daily demand for skis is given in the table.\n$\\begin{array}{cccccc}\\hline Probability & 0.2 & 0.175 & 0.25 & 0.2 & 0.175 \\\\ \\hline Number of Customers & 70 & 71 & 72 & 73 & 74 \\\\ \\hline \\end{array}$\na) Find the expected number of customers: [ANS]\nb) If 70 sets of skis are available, compute Nick's expected profit: [ANS]\nc) If 71 sets of skis are available, compute Nick's expected profit: [ANS]\nd) If 72 sets of skis are available, compute Nick's expected profit: [ANS]\ne) If 73 sets of skis are available, compute Nick's expected profit: [ANS]\nf) If 74 sets of skis are available, compute Nick's expected profit: [ANS]\ng) How many sets of skis should Nick have ready for rental to maximize expected profit? [ANS]", "answer_v1": [ "71.975", "1120", "1131.2", "1138.2", "1139.2", "1135.4", "73" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Nick's Ski Rental rents skis, boots, and poles for \\$ 17 per day. The daily cost per set of skiis is \\$ 6. It includes maintenance, storage, and overhead. Daily profits depend on daily demand for skis and the number of sets available. Nick knows that on a typical weekend the daily demand for skis is given in the table.\n$\\begin{array}{cccccc}\\hline Probability & 0.075 & 0.225 & 0.475 & 0.125 & 0.1 \\\\ \\hline Number of Customers & 90 & 91 & 92 & 93 & 94 \\\\ \\hline \\end{array}$\na) Find the expected number of customers: [ANS]\nb) If 90 sets of skis are available, compute Nick's expected profit: [ANS]\nc) If 91 sets of skis are available, compute Nick's expected profit: [ANS]\nd) If 92 sets of skis are available, compute Nick's expected profit: [ANS]\ne) If 93 sets of skis are available, compute Nick's expected profit: [ANS]\nf) If 94 sets of skis are available, compute Nick's expected profit: [ANS]\ng) How many sets of skis should Nick have ready for rental to maximize expected profit? [ANS]", "answer_v2": [ "91.95", "990", "999.725", "1005.625", "1003.45", "999.15", "92" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Mike's Ski Rental rents skis, boots, and poles for \\$ 27 per day. The daily cost per set of skiis is \\$ 10. It includes maintenance, storage, and overhead. Daily profits depend on daily demand for skis and the number of sets available. Mike knows that on a typical weekend the daily demand for skis is given in the table.\n$\\begin{array}{cccccc}\\hline Probability & 0.125 & 0.175 & 0.45 & 0.15 & 0.1 \\\\ \\hline Number of Customers & 65 & 66 & 67 & 68 & 69 \\\\ \\hline \\end{array}$\na) Find the expected number of customers: [ANS]\nb) If 65 sets of skis are available, compute Mike's expected profit: [ANS]\nc) If 66 sets of skis are available, compute Mike's expected profit: [ANS]\nd) If 67 sets of skis are available, compute Mike's expected profit: [ANS]\ne) If 68 sets of skis are available, compute Mike's expected profit: [ANS]\nf) If 69 sets of skis are available, compute Mike's expected profit: [ANS]\ng) How many sets of skis should Mike have ready for rental to maximize expected profit? [ANS]", "answer_v3": [ "66.925", "1105", "1118.625", "1127.525", "1124.275", "1116.975", "67" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Probability_0152", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "probability" ], "problem_v1": "To determine whether or not they have a certain disease, 272 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group the people in groups of 16. The blood samples of the 16 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 16 people (we are assuming that the pooled test will be positive if and only if at least one person in the pool has the desease); whereas, if the test is positive each of the 16 people will also be individually tested and, in all, 17 tests will be made on this group. Assume the probability that a person has the desease is 0.07 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 272 people. answer: [ANS]", "answer_v1": [ "203.828149935424" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "To determine whether or not they have a certain disease, 120 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group the people in groups of 20. The blood samples of the 20 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 20 people (we are assuming that the pooled test will be positive if and only if at least one person in the pool has the desease); whereas, if the test is positive each of the 20 people will also be individually tested and, in all, 21 tests will be made on this group. Assume the probability that a person has the desease is 0.03 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 120 people. answer: [ANS]", "answer_v2": [ "60.7446788487904" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "To determine whether or not they have a certain disease, 160 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group the people in groups of 16. The blood samples of the 16 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 16 people (we are assuming that the pooled test will be positive if and only if at least one person in the pool has the desease); whereas, if the test is positive each of the 16 people will also be individually tested and, in all, 17 tests will be made on this group. Assume the probability that a person has the desease is 0.04 for all people, independently of each other, and compute the expected number of tests necessary for the entire group of 160 people. answer: [ANS]", "answer_v3": [ "86.7355320533644" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0153", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "probability", "Expected Value" ], "problem_v1": "A charity holds a raffle in which each ticket is sold for \\$40. A total of 11000 tickets are sold. They raffle one grand prize which is a Mercedes Benz E350 valued at \\$60000 along with 3 second prizes of Kawasaki motorcycles valued at \\$12000 each. What are the expected winnings for a single ticket buyer? Express to at least three decimal place accuracy in dollar form (as opposed to cents). Answer: \\$ [ANS]", "answer_v1": [ "-31.2727272727273" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A charity holds a raffle in which each ticket is sold for \\$25. A total of 9400 tickets are sold. They raffle one grand prize which is a Mercedes Benz E350 valued at \\$45000 along with 6 second prizes of Ducati motorcycles valued at \\$15000 each. What are the expected winnings for a single ticket buyer? Express to at least three decimal place accuracy in dollar form (as opposed to cents). Answer: \\$ [ANS]", "answer_v2": [ "-10.6382978723404" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A charity holds a raffle in which each ticket is sold for \\$25. A total of 10200 tickets are sold. They raffle one grand prize which is a BMW M3 valued at \\$50000 along with 3 second prizes of Honda motorcycles valued at \\$12000 each. What are the expected winnings for a single ticket buyer? Express to at least three decimal place accuracy in dollar form (as opposed to cents). Answer: \\$ [ANS]", "answer_v3": [ "-16.5686274509804" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0154", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "probability" ], "problem_v1": "A \\$85000 oil detector is lowered under the sea to detect oil fields, and it becomes detached from the ship. If the instrument is not found within 36 hours, it will crack under the pressure of the sea. It is assumed that a SCUBA diver will find it with probability 0.86, but it costs \\$ 500 to hire each diver.\na)How many SCUBA divers should be hired in order to maximize the expected gain? answer: (round to nearest integer) [ANS]\nb)Using your answer to part a), what is the expected gain? answer: \\$ [ANS]", "answer_v1": [ "3", "83266.76" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A \\$60000 oil detector is lowered under the sea to detect oil fields, and it becomes detached from the ship. If the instrument is not found within 36 hours, it will crack under the pressure of the sea. It is assumed that a SCUBA diver will find it with probability 0.73, but it costs \\$ 700 to hire each diver.\na)How many SCUBA divers should be hired in order to maximize the expected gain? answer: (round to nearest integer) [ANS]\nb)Using your answer to part a), what is the expected gain? answer: \\$ [ANS]", "answer_v2": [ "4", "56881.1354" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A \\$70000 oil detector is lowered under the sea to detect oil fields, and it becomes detached from the ship. If the instrument is not found within 36 hours, it will crack under the pressure of the sea. It is assumed that a SCUBA diver will find it with probability 0.77, but it costs \\$ 500 to hire each diver.\na)How many SCUBA divers should be hired in order to maximize the expected gain? answer: (round to nearest integer) [ANS]\nb)Using your answer to part a), what is the expected gain? answer: \\$ [ANS]", "answer_v3": [ "4", "67804.1113" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0155", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "probability", "expected value", "random variable" ], "problem_v1": "Sam is applying for a single year life insurance policy worth $\\$54{,}100.00$. If the actuarial tables determine that she will survive the next year with probability $0.996$, what is her expected value for the life insurance policy if the premium is $\\$417.00$? [ANS]\n(Note: Include dollar signs in your answer)", "answer_v1": [ "-200.60" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sam is applying for a single year life insurance policy worth $\\$71{,}600.00$. If the actuarial tables determine that he will survive the next year with probability $0.991$, what is his expected value for the life insurance policy if the premium is $\\$370.00$? [ANS]\n(Note: Include dollar signs in your answer)", "answer_v2": [ "274.40" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sam is applying for a single year life insurance policy worth $\\$55{,}300.00$. If the actuarial tables determine that he will survive the next year with probability $0.992$, what is his expected value for the life insurance policy if the premium is $\\$396.00$? [ANS]\n(Note: Include dollar signs in your answer)", "answer_v3": [ "46.40" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0156", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "probability", "expected value", "random variable" ], "problem_v1": "Find the expected value for the random variable:\n$\\begin{array}{ccccc}\\hline X & 3 & 5 & 7 & 9 \\\\ \\hline P(X) & 0.16 & 0.16 & 0.21 & 0.47 \\\\ \\hline \\end{array}$\n$E(X)=$ [ANS]", "answer_v1": [ "6.98" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the expected value for the random variable:\n$\\begin{array}{ccccc}\\hline X & 1 & 3 & 4 & 5 \\\\ \\hline P(X) & 0.28 & 0.16 & 0.13 & 0.43 \\\\ \\hline \\end{array}$\n$E(X)=$ [ANS]", "answer_v2": [ "3.43" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the expected value for the random variable:\n$\\begin{array}{ccccc}\\hline X & 1 & 3 & 4 & 6 \\\\ \\hline P(X) & 0.14 & 0.16 & 0.26 & 0.44 \\\\ \\hline \\end{array}$\n$E(X)=$ [ANS]", "answer_v3": [ "4.3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0157", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "probability", "expected value", "random variable" ], "problem_v1": "Suppose that from a standard deck, you draw three cards without replacement. What is the expected number of black that you will draw? [ANS]", "answer_v1": [ "1.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that from a standard deck, you draw three cards without replacement. What is the expected number of spades that you will draw? [ANS]", "answer_v2": [ "0.75" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that from a standard deck, you draw three cards without replacement. What is the expected number of kings that you will draw? [ANS]", "answer_v3": [ "0.230769" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0158", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "algebra", "probability", "expected value" ], "problem_v1": "Two dice are tossed $450$ times. How many times would you expect to get a double? Answer: [ANS]", "answer_v1": [ "1/6*450" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two dice are tossed $150$ times. How many times would you expect to get a double? Answer: [ANS]", "answer_v2": [ "1/6*150" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two dice are tossed $240$ times. How many times would you expect to get a double? Answer: [ANS]", "answer_v3": [ "1/6*240" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0159", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "3", "keywords": [ "Random variable", "discrete variables", "expectation", "finding expectation of a discrete variable." ], "problem_v1": "The discrete random variable X has a probability distribution defined below:\n$\\begin{array}{ccccc}\\hline x & 0 & 2 & 4 & 8 \\\\ \\hline P(X=x) & 1/4 & 1/8 & 1/2 & 1/8 \\\\ \\hline \\end{array}$\nThe expectation of X is: [ANS] A. 13/4 B. 7/2 C. 2/3 D. 14 E. 3", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The discrete random variable X has a probability distribution defined below:\n$\\begin{array}{ccccc}\\hline x & 0 & 2 & 4 & 8 \\\\ \\hline P(X=x) & 1/4 & 1/8 & 1/2 & 1/8 \\\\ \\hline \\end{array}$\nThe expectation of X is: [ANS] A. 7/2 B. 2/3 C. 14 D. 13/4 E. 3", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The discrete random variable X has a probability distribution defined below:\n$\\begin{array}{ccccc}\\hline x & 0 & 2 & 4 & 8 \\\\ \\hline P(X=x) & 1/4 & 1/8 & 1/2 & 1/8 \\\\ \\hline \\end{array}$\nThe expectation of X is: [ANS] A. 3 B. 7/2 C. 14 D. 2/3 E. 13/4", "answer_v3": [ "E" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0160", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "probability", "expected value" ], "problem_v1": "The owner of a small firm has just purchased a personal computer, which she expects will serve her for the next two years. The owner has been told that she \"must\" buy a surge suppressor to provide protection for her new hardware against possible surges or variations in the electrical current, which have the capacity to damage the computer. The amount of damage to the computer depends on the strength of the surge. It has been estimated that there is a 3\\% chance of incurring 550 dollar damage, 5\\% chance of incurring 150 dollar damage, and 13\\% chance of incurring 100 dollar damage from a surge within the next two years. An inexpensive suppressor, which would provide protection for only one surge, can be purchased. How much should the owner be willing to pay if she makes decisions on the basis of expected value? Expected value=[ANS]", "answer_v1": [ "37" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The owner of a small firm has just purchased a personal computer, which she expects will serve her for the next two years. The owner has been told that she \"must\" buy a surge suppressor to provide protection for her new hardware against possible surges or variations in the electrical current, which have the capacity to damage the computer. The amount of damage to the computer depends on the strength of the surge. It has been estimated that there is a 1\\% chance of incurring 400 dollar damage, 6\\% chance of incurring 300 dollar damage, and 10\\% chance of incurring 100 dollar damage from a surge within the next two years. An inexpensive suppressor, which would provide protection for only one surge, can be purchased. How much should the owner be willing to pay if she makes decisions on the basis of expected value? Expected value=[ANS]", "answer_v2": [ "32" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The owner of a small firm has just purchased a personal computer, which she expects will serve her for the next two years. The owner has been told that she \"must\" buy a surge suppressor to provide protection for her new hardware against possible surges or variations in the electrical current, which have the capacity to damage the computer. The amount of damage to the computer depends on the strength of the surge. It has been estimated that there is a 1\\% chance of incurring 450 dollar damage, 5\\% chance of incurring 150 dollar damage, and 11\\% chance of incurring 100 dollar damage from a surge within the next two years. An inexpensive suppressor, which would provide protection for only one surge, can be purchased. How much should the owner be willing to pay if she makes decisions on the basis of expected value? Expected value=[ANS]", "answer_v3": [ "23" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0161", "subject": "Probability", "topic": "Random variables", "subtopic": "Expectation", "level": "2", "keywords": [ "probability", "expected value" ], "problem_v1": "To examine the effectiveness of its four annual advertising promotions, a mail order company has sent a questionnaire to each of its customers, asking how many of the previous year's promotions prompted orders that would not have otherwise been made. The accompanying table lists the probabilities that were derived from the questionnaire, where X is the random variable representing the number of promotions that prompted orders. If we assume that overall customer behavior next year will be the same as last year, what is the expected number of promotions that each customer will take advantage of next year by ordering goods that otherwise would not be purchased?\n$\\begin{array}{cccccc}\\hline X & 0 & 1 & 2 & 3 & 4 \\\\ \\hline P(X) & 0.088 & 0.229 & 0.362 & 0.186 & 0.135 \\\\ \\hline \\end{array}$\nExpected value=[ANS]\nA previous analysis of historical records found that the mean value of orders for promotional goods is 26 dollars, with the company earning a gross profit of 23\\% on each order. Calculate the expected value of the profit contribution next year. Expected value=[ANS]\nThe fixed cost of conducting the four promotions is estimated to be 16000 dollars with a variable cost of 4.25 dollars per customer for mailing and handling costs. What is the minimum number of customers required by the company in order to cover the cost of promotions? (Round your answer to the next highest integer.) Breakeven point=[ANS]", "answer_v1": [ "2.051", "12.26498", "1997" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "To examine the effectiveness of its four annual advertising promotions, a mail order company has sent a questionnaire to each of its customers, asking how many of the previous year's promotions prompted orders that would not have otherwise been made. The accompanying table lists the probabilities that were derived from the questionnaire, where X is the random variable representing the number of promotions that prompted orders. If we assume that overall customer behavior next year will be the same as last year, what is the expected number of promotions that each customer will take advantage of next year by ordering goods that otherwise would not be purchased?\n$\\begin{array}{cccccc}\\hline X & 0 & 1 & 2 & 3 & 4 \\\\ \\hline P(X) & 0.054 & 0.246 & 0.315 & 0.167 & 0.218 \\\\ \\hline \\end{array}$\nExpected value=[ANS]\nA previous analysis of historical records found that the mean value of orders for promotional goods is 39 dollars, with the company earning a gross profit of 23\\% on each order. Calculate the expected value of the profit contribution next year. Expected value=[ANS]\nThe fixed cost of conducting the four promotions is estimated to be 12000 dollars with a variable cost of 3.25 dollars per customer for mailing and handling costs. What is the minimum number of customers required by the company in order to cover the cost of promotions? (Round your answer to the next highest integer.) Breakeven point=[ANS]", "answer_v2": [ "2.249", "20.17353", "710" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "To examine the effectiveness of its four annual advertising promotions, a mail order company has sent a questionnaire to each of its customers, asking how many of the previous year's promotions prompted orders that would not have otherwise been made. The accompanying table lists the probabilities that were derived from the questionnaire, where X is the random variable representing the number of promotions that prompted orders. If we assume that overall customer behavior next year will be the same as last year, what is the expected number of promotions that each customer will take advantage of next year by ordering goods that otherwise would not be purchased?\n$\\begin{array}{cccccc}\\hline X & 0 & 1 & 2 & 3 & 4 \\\\ \\hline P(X) & 0.065 & 0.23 & 0.328 & 0.177 & 0.2 \\\\ \\hline \\end{array}$\nExpected value=[ANS]\nA previous analysis of historical records found that the mean value of orders for promotional goods is 24 dollars, with the company earning a gross profit of 23\\% on each order. Calculate the expected value of the profit contribution next year. Expected value=[ANS]\nThe fixed cost of conducting the four promotions is estimated to be 18000 dollars with a variable cost of 5.75 dollars per customer for mailing and handling costs. What is the minimum number of customers required by the company in order to cover the cost of promotions? (Round your answer to the next highest integer.) Breakeven point=[ANS]", "answer_v3": [ "2.217", "12.23784", "2775" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0162", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "Random Variable", "Mean", "Standard Deviation" ], "problem_v1": "The mean and standard deviation of a random variable $x$ are $6$ and $3$ respectively. Find the mean and standard deviation of the given random variables: (1) $\\ y=x+6$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]\n(2) $\\ v=7x$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]\n(3) $\\ w=7x+6$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]", "answer_v1": [ "12", "3", "42", "21", "48", "21" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The mean and standard deviation of a random variable $x$ are $-10$ and $4$ respectively. Find the mean and standard deviation of the given random variables: (1) $\\ y=x+2$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]\n(2) $\\ v=4x$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]\n(3) $\\ w=4x+2$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]", "answer_v2": [ "-8", "4", "-40", "16", "-38", "16" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The mean and standard deviation of a random variable $x$ are $-4$ and $3$ respectively. Find the mean and standard deviation of the given random variables: (1) $\\ y=x+3$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]\n(2) $\\ v=6x$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]\n(3) $\\ w=6x+3$ $\\mu=$ [ANS]\n$\\sigma=$ [ANS]", "answer_v3": [ "-1", "3", "-24", "18", "-21", "18" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0163", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "Random Variable" ], "problem_v1": "Prizes and the chances of winning in a sweepstakes are given in the table below. $\\begin{array}{cc}\\hline Prize & Chances \\\\ \\hline \\$25,000,000 & 1 chance in 300,000,000 \\\\ \\hline \\$450,000 & 1 chance in 200,000,000 \\\\ \\hline \\$25,000 & 1 chance in 30,000,000 \\\\ \\hline \\$15,000 & 1 chance in 3,000,000 \\\\ \\hline \\$400 & 1 chance in 500,000 \\\\ \\hline A watch valued at \\$80 & 1 chance in 4,000 \\\\ \\hline \\end{array}$\n(a) Find the expected value (in dollars) of the amount won by one entry. [ANS]\n(b) Find the expected value (in dollars) if the cost of entering this sweepstakes is the cost of a postage stamp (34 cents) [ANS]", "answer_v1": [ "0.112216666666667", "-0.227783333333333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Prizes and the chances of winning in a sweepstakes are given in the table below. $\\begin{array}{cc}\\hline Prize & Chances \\\\ \\hline \\$10,000,000 & 1 chance in 400,000,000 \\\\ \\hline \\$150,000 & 1 chance in 150,000,000 \\\\ \\hline \\$75,000 & 1 chance in 30,000,000 \\\\ \\hline \\$5,000 & 1 chance in 2,000,000 \\\\ \\hline \\$600 & 1 chance in 100,000 \\\\ \\hline A watch valued at \\$80 & 1 chance in 6,000 \\\\ \\hline \\end{array}$\n(a) Find the expected value (in dollars) of the amount won by one entry. [ANS]\n(b) Find the expected value (in dollars) if the cost of entering this sweepstakes is the cost of a postage stamp (34 cents) [ANS]", "answer_v2": [ "0.0503333333333333", "-0.289666666666667" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Prizes and the chances of winning in a sweepstakes are given in the table below. $\\begin{array}{cc}\\hline Prize & Chances \\\\ \\hline \\$15,000,000 & 1 chance in 300,000,000 \\\\ \\hline \\$250,000 & 1 chance in 150,000,000 \\\\ \\hline \\$25,000 & 1 chance in 40,000,000 \\\\ \\hline \\$20,000 & 1 chance in 5,000,000 \\\\ \\hline \\$800 & 1 chance in 200,000 \\\\ \\hline A watch valued at \\$60 & 1 chance in 4,000 \\\\ \\hline \\end{array}$\n(a) Find the expected value (in dollars) of the amount won by one entry. [ANS]\n(b) Find the expected value (in dollars) if the cost of entering this sweepstakes is the cost of a postage stamp (34 cents) [ANS]", "answer_v3": [ "0.0752916666666667", "-0.264708333333333" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0164", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "Expected Value", "Random Variable", "Variance" ], "problem_v1": "If $E[X]=3$ and $\\mbox{Var}(X)=4$, then $E[(4+5 X)^2]=$ [ANS]\nand $\\mbox{Var}(2+3 X)=$ [ANS].", "answer_v1": [ "461", "36" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $E[X]=-5$ and $\\mbox{Var}(X)=5$, then $E[(1+3 X)^2]=$ [ANS]\nand $\\mbox{Var}(5+3 X)=$ [ANS].", "answer_v2": [ "241", "45" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $E[X]=-2$ and $\\mbox{Var}(X)=4$, then $E[(2+4 X)^2]=$ [ANS]\nand $\\mbox{Var}(2+3 X)=$ [ANS].", "answer_v3": [ "100", "36" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0165", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "central limit" ], "problem_v1": "A poll of 88 students found that 39\\% were in favor of raising tution to build a new math building. The standard deviation of this poll is 7\\%. What would be the standard deviation if the sample size were increased from 88 to 248? Answer: [ANS] \\%", "answer_v1": [ "4.16978378026889" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A poll of 54 students found that 24\\% were in favor of raising tution to build a new football stadium. The standard deviation of this poll is 5\\%. What would be the standard deviation if the sample size were increased from 54 to 287? Answer: [ANS] \\%", "answer_v2": [ "2.16883211720346" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A poll of 65 students found that 28\\% were in favor of raising tution to pave new parking lots. The standard deviation of this poll is 6\\%. What would be the standard deviation if the sample size were increased from 65 to 233? Answer: [ANS] \\%", "answer_v3": [ "3.1690563981942" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0166", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "3", "keywords": [ "statistics", "variance" ], "problem_v1": "Let $V$ be a random variable be mean $\\mu=5$ and variance $\\sigma^2=5$. Part a) What is $E(3 V+5)$? [ANS]\nPart b) What is $Var(3 V+5)$? [ANS]\nPart c) What is $Var(3 V-5)$? [ANS]", "answer_v1": [ "20", "45", "45" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "MathJax.Hub.Config({MathMenu: {showContext: true}}); if(!window.MathJax) (function () {var script=document.createElement(\"script\"); script.type=\"text/javascript\"; script.src=\"https://work2.classviva.com/webwork2_files/mathjax/MathJax.js?config=TeX-MML-AM_HTMLorMML-full\"; document.getElementsByTagName(\"head\")[0].appendChild(script);})(); Let $V$ be a random variable be mean $\\mu=1$ and variance $\\sigma^2=6$. Part a) What is $E(1 V+4)$? [ANS]\nPart b) What is $Var(1 V+4)$? [ANS]\nPart c) What is $Var(1 V-4)$? [ANS]", "answer_v2": [ "5", "6", "6" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "MathJax.Hub.Config({MathMenu: {showContext: true}}); if(!window.MathJax) (function () {var script=document.createElement(\"script\"); script.type=\"text/javascript\"; script.src=\"https://work2.classviva.com/webwork2_files/mathjax/MathJax.js?config=TeX-MML-AM_HTMLorMML-full\"; document.getElementsByTagName(\"head\")[0].appendChild(script);})(); Let $V$ be a random variable be mean $\\mu=2$ and variance $\\sigma^2=5$. Part a) What is $E(2 V+5)$? [ANS]\nPart b) What is $Var(2 V+5)$? [ANS]\nPart c) What is $Var(2 V-5)$? [ANS]", "answer_v3": [ "9", "20", "20" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0167", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "3", "keywords": [ "Random variables", "discrete variable", "discrete Uniform distribution", "expectation", "variance", "finding mass function", "expectation and variance for a discrete Uniform distribution" ], "problem_v1": "You may make reference to the following three results in this question: $\\sum_{i=m}^n c=(n-m+1) c$ $\\sum_{i=1}^n i=\\frac{1}{2} n(n+1)$ $\\sum_{i=1}^n i^2=\\frac{1}{6} n(n+1)(2n+1).$\nThe random variable $X$ has the discrete Uniform distribution given by $P(X=j)=k$ for $j=2, 3, 4,..., 31$. Find (to 3 d.p.):\nPart a) The value of $k$ given as a fraction. [ANS]\nPart b) The expectation of $X$ given to three decimal places. [ANS]\nPart c) The variance of $X$ given to three decimal places. [ANS]", "answer_v1": [ "0.0333333", "16.5", "74.917" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "You may make reference to the following three results in this question: $\\sum_{i=m}^n c=(n-m+1) c$ $\\sum_{i=1}^n i=\\frac{1}{2} n(n+1)$ $\\sum_{i=1}^n i^2=\\frac{1}{6} n(n+1)(2n+1).$\nThe random variable $X$ has the discrete Uniform distribution given by $P(X=j)=k$ for $j=2, 3, 4,..., 28$. Find (to 3 d.p.):\nPart a) The value of $k$ given as a fraction. [ANS]\nPart b) The expectation of $X$ given to three decimal places. [ANS]\nPart c) The variance of $X$ given to three decimal places. [ANS]", "answer_v2": [ "0.037037", "15", "60.667" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "You may make reference to the following three results in this question: $\\sum_{i=m}^n c=(n-m+1) c$ $\\sum_{i=1}^n i=\\frac{1}{2} n(n+1)$ $\\sum_{i=1}^n i^2=\\frac{1}{6} n(n+1)(2n+1).$\nThe random variable $X$ has the discrete Uniform distribution given by $P(X=j)=k$ for $j=2, 3, 4,..., 29$. Find (to 3 d.p.):\nPart a) The value of $k$ given as a fraction. [ANS]\nPart b) The expectation of $X$ given to three decimal places. [ANS]\nPart c) The variance of $X$ given to three decimal places. [ANS]", "answer_v3": [ "0.0357143", "15.5", "65.25" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0168", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "3", "keywords": [ "statistics", "multiple choice", "variables" ], "problem_v1": "The length of a coil of copper wire is a random variable with mean 150 $m$ and standard deviation 8 $m$.\nIf we choose five coils of wire at random, what is the variance of the total length of the wire in the coils? [ANS] A. $320m^{2}$ B. $0.8m^{2}$ C. $1600m^{2}$ D. $40m^{2}$ E. $200m^{2}$", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The length of a coil of copper wire is a random variable with mean 150 $m$ and standard deviation 2.5 $m$.\nIf we choose five coils of wire at random, what is the variance of the total length of the wire in the coils? [ANS] A. $0.8m^{2}$ B. $156.25m^{2}$ C. $12.5m^{2}$ D. $31.25m^{2}$ E. $62.5m^{2}$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The length of a coil of copper wire is a random variable with mean 150 $m$ and standard deviation 4.5 $m$.\nIf we choose five coils of wire at random, what is the variance of the total length of the wire in the coils? [ANS] A. $112.5m^{2}$ B. $0.8m^{2}$ C. $22.5m^{2}$ D. $506.25m^{2}$ E. $101.25m^{2}$", "answer_v3": [ "E" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0169", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "Summary statistics; mean", "standard deviation; linear transformation of data." ], "problem_v1": "A North American company manufactures rivets for use in car production. A sample of forty rivets from the production line had mean 6.877 and standard deviation 0.216 (both in 1/100 of an inch). On communicating these results to the company headquarters in Europe, a request is made for the summary statistics to be converted into millimeters. Given that one inch is 2.538 centimeters, find the mean and standard deviation of the sample in millimeters. What is the mean in millimeters? (Provide your answer to 3 significant figures.) [ANS]\nWhat is the standard deviation in millimeters? (Provide your answer to 3 significant figures.) [ANS]", "answer_v1": [ "1.74538", "0.0548208" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A North American company manufactures rivets for use in car production. A sample of forty rivets from the production line had mean 6.541 and standard deviation 0.286 (both in 1/100 of an inch). On communicating these results to the company headquarters in Europe, a request is made for the summary statistics to be converted into millimeters. Given that one inch is 2.538 centimeters, find the mean and standard deviation of the sample in millimeters. What is the mean in millimeters? (Provide your answer to 3 significant figures.) [ANS]\nWhat is the standard deviation in millimeters? (Provide your answer to 3 significant figures.) [ANS]", "answer_v2": [ "1.66011", "0.0725868" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A North American company manufactures rivets for use in car production. A sample of forty rivets from the production line had mean 6.657 and standard deviation 0.221 (both in 1/100 of an inch). On communicating these results to the company headquarters in Europe, a request is made for the summary statistics to be converted into millimeters. Given that one inch is 2.538 centimeters, find the mean and standard deviation of the sample in millimeters. What is the mean in millimeters? (Provide your answer to 3 significant figures.) [ANS]\nWhat is the standard deviation in millimeters? (Provide your answer to 3 significant figures.) [ANS]", "answer_v3": [ "1.68955", "0.0560898" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0170", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "statistics", "probability", "bivariate random variable" ], "problem_v1": "There are four activities along the critical path for a project. The expected values and variances of the completion times of the activities are listed below. Determine the expected value and variance of the completion time of the project.\n\\begin{array}{c|c|c} \\mbox{Activity} & \\mbox{Expected Completion Time (Days)} & \\mbox{Variance} \\\\ \\hline\\hline 1 & 18 & 7 \\\\ \\hline 2 & 11 & 5 \\\\ \\hline 3 & 25 & 4 \\\\ \\hline 4 & 7 & 2 \\\\ \\hline \\end{array}\nExpected value of completion time of project=[ANS]\nVariance of completion time of project=[ANS]", "answer_v1": [ "61", "18" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "There are four activities along the critical path for a project. The expected values and variances of the completion times of the activities are listed below. Determine the expected value and variance of the completion time of the project.\n\\begin{array}{c|c|c} \\mbox{Activity} & \\mbox{Expected Completion Time (Days)} & \\mbox{Variance} \\\\ \\hline\\hline 1 & 16 & 8 \\\\ \\hline 2 & 10 & 4 \\\\ \\hline 3 & 27 & 4 \\\\ \\hline 4 & 5 & 1 \\\\ \\hline \\end{array}\nExpected value of completion time of project=[ANS]\nVariance of completion time of project=[ANS]", "answer_v2": [ "58", "17" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "There are four activities along the critical path for a project. The expected values and variances of the completion times of the activities are listed below. Determine the expected value and variance of the completion time of the project.\n\\begin{array}{c|c|c} \\mbox{Activity} & \\mbox{Expected Completion Time (Days)} & \\mbox{Variance} \\\\ \\hline\\hline 1 & 16 & 7 \\\\ \\hline 2 & 10 & 4 \\\\ \\hline 3 & 24 & 5 \\\\ \\hline 4 & 8 & 2 \\\\ \\hline \\end{array}\nExpected value of completion time of project=[ANS]\nVariance of completion time of project=[ANS]", "answer_v3": [ "58", "18" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0171", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "probability", "mean", "standard deviation" ], "problem_v1": "When parking a car in a downtown parking lot, drivers pay according to the number of hours or fraction thereof. The probability distribution of the number of hours cars are parked has been estimated as follows:\n$\\begin{array}{ccccccccc}\\hline X & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline P(X) & 0.23 & 0.129 & 0.119 & 0.093 & 0.056 & 0.026 & 0.031 & 0.316 \\\\ \\hline \\end{array}$\nA.\nMean=[ANS]\nB. Standard Deviation=[ANS]\nThe cost of parking is 3.75 dollars per hour. Calculate the mean and standard deviation of the amount of revenue each car generates. A.\nMean=[ANS]\nB. Standard Deviation=[ANS]", "answer_v1": [ "4.398", "2.83577079468705", "16.4925", "10.6341404800764" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "When parking a car in a downtown parking lot, drivers pay according to the number of hours or fraction thereof. The probability distribution of the number of hours cars are parked has been estimated as follows:\n$\\begin{array}{ccccccccc}\\hline X & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline P(X) & 0.203 & 0.146 & 0.104 & 0.083 & 0.069 & 0.026 & 0.023 & 0.346 \\\\ \\hline \\end{array}$\nA.\nMean=[ANS]\nB. Standard Deviation=[ANS]\nThe cost of parking is 3 dollars per hour. Calculate the mean and standard deviation of the amount of revenue each car generates. A.\nMean=[ANS]\nB. Standard Deviation=[ANS]", "answer_v2": [ "4.569", "2.84732137279935", "13.707", "8.54196411839806" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "When parking a car in a downtown parking lot, drivers pay according to the number of hours or fraction thereof. The probability distribution of the number of hours cars are parked has been estimated as follows:\n$\\begin{array}{ccccccccc}\\hline X & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline P(X) & 0.212 & 0.13 & 0.108 & 0.089 & 0.054 & 0.027 & 0.036 & 0.344 \\\\ \\hline \\end{array}$\nA.\nMean=[ANS]\nB. Standard Deviation=[ANS]\nThe cost of parking is 4.75 dollars per hour. Calculate the mean and standard deviation of the amount of revenue each car generates. A.\nMean=[ANS]\nB. Standard Deviation=[ANS]", "answer_v3": [ "4.588", "2.86011468301535", "21.793", "13.5855447443229" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0172", "subject": "Probability", "topic": "Random variables", "subtopic": "Variance, standard deviation", "level": "2", "keywords": [ "probability", "mean", "variance", "standard deviation" ], "problem_v1": "Find the mean, variance and standard deviation for the probability distribution given below:\n$\\begin{array}{ccccc}\\hline X &-1 & 3 & 9 & 10 \\\\ \\hline P(X) & 0.587 & 0.129 & 0.215 & 0.069 \\\\ \\hline \\end{array}$\nA.\nMean=[ANS]\nB. Variance=[ANS]\nC. Standard Deviation=[ANS]", "answer_v1": [ "2.425", "20.182375", "4.49247982744497" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the mean, variance and standard deviation for the probability distribution given below:\n$\\begin{array}{ccccc}\\hline X &-3 & 7 & 9 & 8 \\\\ \\hline P(X) & 0.554 & 0.146 & 0.203 & 0.097 \\\\ \\hline \\end{array}$\nA.\nMean=[ANS]\nB. Variance=[ANS]\nC. Standard Deviation=[ANS]", "answer_v2": [ "1.963", "30.937631", "5.56216064133355" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the mean, variance and standard deviation for the probability distribution given below:\n$\\begin{array}{ccccc}\\hline X &-2 & 3 & 9 & 12 \\\\ \\hline P(X) & 0.565 & 0.13 & 0.206 & 0.099 \\\\ \\hline \\end{array}$\nA.\nMean=[ANS]\nB. Variance=[ANS]\nC. Standard Deviation=[ANS]", "answer_v3": [ "2.302", "29.072796", "5.39191950978499" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0173", "subject": "Probability", "topic": "Random variables", "subtopic": "Discrete: probability mass function", "level": "2", "keywords": [ "Random Variable" ], "problem_v1": "Let $X$ represent the difference between the number of heads and the number of tails when a coin is tossed $43$ times. Then $P(X=9)=$ [ANS]", "answer_v1": [ "0.0957633457705924" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $X$ represent the difference between the number of heads and the number of tails when a coin is tossed $50$ times. Then $P(X=2)=$ [ANS]", "answer_v2": [ "0.215913793575417" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $X$ represent the difference between the number of heads and the number of tails when a coin is tossed $42$ times. Then $P(X=4)=$ [ANS]", "answer_v3": [ "0.203169889027777" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0174", "subject": "Probability", "topic": "Random variables", "subtopic": "Discrete: probability mass function", "level": "2", "keywords": [ "Random Variable", "Probability Density Function", "PDF" ], "problem_v1": "Two fair dice are rolled $5$ times. Let the random variable $x$ represent the number of times that the sum $10$ occurs. The table below describes the probability distribution. Find the value of the missing probability.\n$\\begin{array}{cc}\\hline x & P(x) \\\\ \\hline 0 & 0.64722784850823 \\\\ \\hline 1 & 0.29419447659465 \\\\ \\hline 2 & 0.0534899048353909 \\\\ \\hline 3 & [ANS] \\\\ \\hline 4 & 0.000221032664609053 \\\\ \\hline 5 & 4.01877572016461e-06 \\\\ \\hline \\end{array}$ Would it be unusual to roll a pair of dice $5$ times and get no $10$ s? (enter YES or NO) $\\ $ [ANS]", "answer_v1": [ "0.00486271862139918", "NO" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "Two fair dice are rolled $5$ times. Let the random variable $x$ represent the number of times that the sum $2$ occurs. The table below describes the probability distribution. Find the value of the missing probability.\n$\\begin{array}{cc}\\hline x & P(x) \\\\ \\hline 0 & 0.868615786121484 \\\\ \\hline 1 & 0.124087969445926 \\\\ \\hline 2 & 0.00709074111119579 \\\\ \\hline 3 & 0.000202592603177022 \\\\ \\hline 4 & [ANS] \\\\ \\hline 5 & 1.65381716879202e-08 \\\\ \\hline \\end{array}$ Would it be unusual to roll a pair of dice $5$ times and get no $2$ s? (enter YES or NO) $\\ $ [ANS]", "answer_v2": [ "2.89418004538603E-06", "NO" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "Two fair dice are rolled $5$ times. Let the random variable $x$ represent the number of times that the sum $8$ occurs. The table below describes the probability distribution. Find the value of the missing probability.\n$\\begin{array}{cc}\\hline x & P(x) \\\\ \\hline 0 & 0.473473814517392 \\\\ \\hline 1 & 0.381833721384994 \\\\ \\hline 2 & [ANS] \\\\ \\hline 3 & 0.0198664787401142 \\\\ \\hline 4 & 0.00160213538226727 \\\\ \\hline 5 & 5.16817865247506e-05 \\\\ \\hline \\end{array}$ Would it be unusual to roll a pair of dice $5$ times and get no $8$ s? (enter YES or NO) $\\ $ [ANS]", "answer_v3": [ "0.123172168188708", "NO" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Probability_0175", "subject": "Probability", "topic": "Random variables", "subtopic": "Discrete: probability mass function", "level": "2", "keywords": [ "statistics", "probability" ], "problem_v1": "Let the random variable $X$ be the number of rooms in a randomly chosen owner-occupied housing unit in a certain city. The distribution for the units is given below.\n$\\begin{array}{ccccccccc}\\hline X & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\ \\hline P(X) & 0.09 & 0.25 & 0.36 & 0.18 & 0.05 & 0.03 & 0.02 &? \\\\ \\hline \\end{array}$\n(a) Is $X$ a discrete or continuous random variable? (Type: DISCRETE or CONTINUOUS) ANSWER: [ANS]\n(b) What must be the probability of choosing a unit with 10 rooms? $P(X=10)$=[ANS]\n(c) What is the probability that a unit chosen at random is not a 10-room unit? $P(X \\neq 10)$=[ANS]\n(d) What is the probability that a unit chosen at random has less than five rooms? $P(X<5)$=[ANS]\n(e) What is the probability that a unit chosen at random has more than 5 rooms? $P(X > 5)$=[ANS]", "answer_v1": [ "DISCRETE", "0.02", "0.98", "0.34", "0.3" ], "answer_type_v1": [ "MCS", "NV", "NV", "NV", "NV" ], "options_v1": [ [ "DISCRETE", "CONTINUOUS" ], [], [], [], [] ], "problem_v2": "Let the random variable $X$ be the number of rooms in a randomly chosen owner-occupied housing unit in a certain city. The distribution for the units is given below.\n$\\begin{array}{ccccccccc}\\hline X & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\ \\hline P(X) & 0.05 & 0.29 & 0.37 & 0.15 & 0.04 & 0.05 & 0.04 &? \\\\ \\hline \\end{array}$\n(a) Is $X$ a discrete or continuous random variable? (Type: DISCRETE or CONTINUOUS) ANSWER: [ANS]\n(b) What must be the probability of choosing a unit with 10 rooms? $P(X=10)$=[ANS]\n(c) What is the probability that a unit chosen at random has less than five rooms? $P(X<5)$=[ANS]\n(d) What is the probability that a unit chosen at random has more than 5 rooms? $P(X > 5)$=[ANS]\n(e) What is the probability that a unit chosen at random has three rooms? $P(X=3)$=[ANS]", "answer_v2": [ "DISCRETE", "0.01", "0.34", "0.29", "0.05" ], "answer_type_v2": [ "MCS", "NV", "NV", "NV", "NV" ], "options_v2": [ [ "DISCRETE", "CONTINUOUS" ], [], [], [], [] ], "problem_v3": "Let the random variable $X$ be the number of rooms in a randomly chosen owner-occupied housing unit in a certain city. The distribution for the units is given below.\n$\\begin{array}{ccccccccc}\\hline X & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\ \\hline P(X) & 0.06 & 0.26 & 0.41 & 0.16 & 0.04 & 0.03 & 0.02 &? \\\\ \\hline \\end{array}$\n(a) Is $X$ a discrete or continuous random variable? (Type: DISCRETE or CONTINUOUS) ANSWER: [ANS]\n(b) What must be the probability of choosing a unit with 10 rooms? $P(X=10)$=[ANS]\n(c) What is the probability that a unit chosen at random has between four and six rooms? $P(4 \\leq X \\leq 6)$=[ANS]\n(d) What is the probability that a unit chosen at random has more than 5 rooms? $P(X > 5)$=[ANS]\n(e) What is the probability that a unit chosen at random has three rooms? $P(X=3)$=[ANS]", "answer_v3": [ "DISCRETE", "0.02", "0.83", "0.27", "0.06" ], "answer_type_v3": [ "MCS", "NV", "NV", "NV", "NV" ], "options_v3": [ [ "DISCRETE", "CONTINUOUS" ], [], [], [], [] ] }, { "id": "Probability_0176", "subject": "Probability", "topic": "Random variables", "subtopic": "Discrete: probability mass function", "level": "2", "keywords": [ "probability", "random sampling", "probability distributions" ], "problem_v1": "$\\begin{array}{cccccc}\\hline x & 20 & 21 & 22 & 23 & 24 \\\\ \\hline F(x) & 18 & 6 & 11 & 4 & 2 \\\\ \\hline \\end{array}$\nLet $x$ be the ages of students in a class. Given the frequency distribution $F(x)$ above, determine the following probabilities:\n(a) $P(x\\ge 23 \\mbox{or} x<21)=$ [ANS]\n(b) $P(x<24)=$ [ANS]\n(c) $P(x\\ge 21)=$ [ANS]", "answer_v1": [ "(4+2+18)/41", "(18+6+11+4)/41", "(6+11+4+2)/41" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "$\\begin{array}{cccccc}\\hline x & 18 & 19 & 20 & 21 & 22 \\\\ \\hline F(x) & 8 & 10 & 7 & 8 & 2 \\\\ \\hline \\end{array}$\nLet $x$ be the ages of students in a class. Given the frequency distribution $F(x)$ above, determine the following probabilities:\n(a) $P(x>19)=$ [ANS]\n(b) $P(x\\le 21)=$ [ANS]\n(c) $P(x<22)=$ [ANS]", "answer_v2": [ "(7+8+2)/35", "(8+10+7+8)/35", "(8+10+7+8)/35" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "$\\begin{array}{cccccc}\\hline x & 18 & 19 & 20 & 21 & 22 \\\\ \\hline F(x) & 10 & 7 & 10 & 3 & 2 \\\\ \\hline \\end{array}$\nLet $x$ be the ages of students in a class. Given the frequency distribution $F(x)$ above, determine the following probabilities:\n(a) $P(x=18)=$ [ANS]\n(b) $P(18\\le x <20)=$ [ANS]\n(c) $P(x<22)=$ [ANS]", "answer_v3": [ "10/32", "(10+7)/32", "(10+7+10+3)/32" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0177", "subject": "Probability", "topic": "Random variables", "subtopic": "Continuous: density function, cumulative distribution function", "level": "4", "keywords": [ "Stochastic processes", "Poisson process", "Exponential distribution", "Gamma distribution", "identify the Exponential distribution as the waiting time between events in a Poisson process", "find probability for an Exponential variable", "identify a Gamma distribution as the sum of waiting times between events in a Poisson process", "recognize a Gamma density from the shape and scale parameters", "find a tail probability for a Gamma distribution" ], "problem_v1": "Let X be Uniformly distributed over the interval $[0, 1]$. Find the density function for $Y=\\sin^{-1} X$. Evaluate the density function (to 2 d.p.) at the value 0.91 radians. [ANS]", "answer_v1": [ "0.61" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let X be an Exponential random variable with mean 1. Find the density function for $Y=\\sqrt{X}$. Evaluate the density function (to 2 d.p.) of Y at the value 1.9. [ANS]", "answer_v2": [ "0.1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let X be an Exponential random variable with mean 1. Find the density function for $Y=\\sqrt{X}$. Evaluate the density function (to 2 d.p.) of Y at the value 1.6. [ANS]", "answer_v3": [ "0.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0178", "subject": "Probability", "topic": "Random variables", "subtopic": "Continuous: density function, cumulative distribution function", "level": "2", "keywords": [ "statistics", "multiple choice", "display methods" ], "problem_v1": "Which of the following is/are always true about a continuous random variable? CHECK ALL THAT APPLY. [ANS] A. The probability that it has a value equal to $x$ is given by $f(x)$ where $f$ is the density function. B. It can only take on integer values. C. The probability that it has a value that falls between real values $a$ and $b$ is given by the area under the density function over the range $(a,b)$. D. None of the above", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Which of the following is/are always true about a continuous random variable? CHECK ALL THAT APPLY. [ANS] A. The probability that it has a value that falls between real values $a$ and $b$ is given by the area under the density function over the range $(a,b)$. B. It can only take on integer values. C. The probability that it has a value equal to $x$ is given by $f(x)$ where $f$ is the density function. D. None of the above", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Which of the following is/are always true about a continuous random variable? CHECK ALL THAT APPLY. [ANS] A. The probability that it has a value equal to $x$ is given by $f(x)$ where $f$ is the density function. B. The probability that it has a value that falls between real values $a$ and $b$ is given by the area under the density function over the range $(a,b)$. C. It can only take on integer values. D. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0179", "subject": "Probability", "topic": "Random variables", "subtopic": "Continuous: density function, cumulative distribution function", "level": "3", "keywords": [ "probability", "Continuous random variable", "Expectation", "Variance", "Computing probability", "Cumulative distribution function", "Probability density function" ], "problem_v1": "The continuous random variable $X$ has a probability density function (pdf) given by f(x)=\\begin{cases} 1-\\frac{1}{2} x & \\mbox{for} 0 \\le x \\le 2 \\\\ 0 & \\mbox{otherwise} \\\\ \\end{cases}\nPart(a) Find the median of $X$, correct to 2 decimal places. [ANS]\nPart(b) Find $E(X)$, correct to 2 decimal places. [ANS]\nPart(c) Two independent observations of $X$ are taken. Find the probability correct to 2 decimal places that one is less than $\\frac{1}{2}$ and the other is greater than $\\frac{1}{2}$. [ANS]\nPart(d) Find Var($X$), correct to 2 decimal places. [ANS]\nPart(e) Find the value of $q$ such that $P(X < q)=\\frac{1}{4}.$ Give your answer as a decimal correct to 3 decimal places. [ANS]\nPart(f) Find $P(X > \\frac{1}{2})$. Give your answer as a decimal, correct to 2 decimal places. [ANS]", "answer_v1": [ "0.59", "0.67", "0.49", "0.22", "0.268", "0.56" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The continuous random variable $X$ has a probability density function (pdf) given by f(x)=\\begin{cases} 1-\\frac{1}{2} x & \\mbox{for} 0 \\le x \\le 2 \\\\ 0 & \\mbox{otherwise} \\\\ \\end{cases}\nPart(a) Find $P(X > \\frac{1}{2})$. Give your answer as a decimal, correct to 2 decimal places. [ANS]\nPart(b) Find $E(\\sqrt{X})$, correct to 2 decimal places. [ANS]\nPart(c) Find the value of $q$ such that $P(X < q)=\\frac{1}{4}.$ Give your answer as a decimal correct to 3 decimal places. [ANS]\nPart(d) Two independent observations of $X$ are taken. Find the probability correct to 2 decimal places that one is less than $\\frac{1}{2}$ and the other is greater than $\\frac{1}{2}$. [ANS]\nPart(e) Find the median of $X$, correct to 2 decimal places. [ANS]\nPart(f) Find the value of $c$ correct to one decimal place given that $E(X+c)=4E(X\u2212c)$. [ANS]", "answer_v2": [ "0.56", "0.75", "0.268", "0.49", "0.59", "0.4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The continuous random variable $X$ has a probability density function (pdf) given by f(x)=\\begin{cases} 1-\\frac{1}{2} x & \\mbox{for} 0 \\le x \\le 2 \\\\ 0 & \\mbox{otherwise} \\\\ \\end{cases}\nPart(a) Find the value of $c$ correct to one decimal place given that $E(X+c)=4E(X\u2212c)$. [ANS]\nPart(b) Find Var($X$), correct to 2 decimal places. [ANS]\nPart(c) Find the value of $q$ such that $P(X < q)=\\frac{1}{4}.$ Give your answer as a decimal correct to 3 decimal places. [ANS]\nPart(d) Find $E(X)$, correct to 2 decimal places. [ANS]\nPart(e) Find $P(X > \\frac{1}{2})$. Give your answer as a decimal, correct to 2 decimal places. [ANS]\nPart(f) Find the median of $X$, correct to 2 decimal places. [ANS]", "answer_v3": [ "0.4", "0.22", "0.268", "0.67", "0.56", "0.59" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0180", "subject": "Probability", "topic": "Random variables", "subtopic": "Generating function", "level": "3", "keywords": [ "moments", "expected value", "expectation", "standard deviation", "variance", "sampling distributions", "binomial distribution", "binomial random variable" ], "problem_v1": "If $Y$ is $binomial(n,p)$, find the MGF of $Y$. $M(t)=$ [ANS]\nIf $n=39$ and $p=0.7$, differentiate the MGF you found above to find the first 3 moments of $Y$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $Y$. $var(Y)=$ [ANS]", "answer_v1": [ "(p*e^t+1-p)^n", "27.3", "753.48", "21013.9", "8.19" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "If $Y$ is $binomial(n,p)$, find the MGF of $Y$. $M(t)=$ [ANS]\nIf $n=57$ and $p=0.1$, differentiate the MGF you found above to find the first 3 moments of $Y$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $Y$. $var(Y)=$ [ANS]", "answer_v2": [ "(p*e^t+1-p)^n", "5.7", "37.62", "277.02", "5.13" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "If $Y$ is $binomial(n,p)$, find the MGF of $Y$. $M(t)=$ [ANS]\nIf $n=40$ and $p=0.3$, differentiate the MGF you found above to find the first 3 moments of $Y$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $Y$. $var(Y)=$ [ANS]", "answer_v3": [ "(p*e^t+1-p)^n", "12", "152.4", "2033.76", "8.4" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0181", "subject": "Probability", "topic": "Random variables", "subtopic": "Generating function", "level": "3", "keywords": [ "moments", "expected value", "expectation", "standard deviation", "variance", "sampling distributions", "poisson distribution" ], "problem_v1": "If $X$ is $poisson(\\lambda)$, find the MGF of $X$. (Enter $\\lambda$ as $l$) $M(t)=$ [ANS]\nIf $\\lambda=16$, differentiate the MGF you found above to find the first 3 moments of $X$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $X$. $var(X)=$ [ANS]", "answer_v1": [ "e^[l*(e^t-1)]", "16", "272", "4880", "16" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "If $X$ is $poisson(\\lambda)$, find the MGF of $X$. (Enter $\\lambda$ as $l$) $M(t)=$ [ANS]\nIf $\\lambda=2$, differentiate the MGF you found above to find the first 3 moments of $X$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $X$. $var(X)=$ [ANS]", "answer_v2": [ "e^[l*(e^t-1)]", "2", "6", "22", "2" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "If $X$ is $poisson(\\lambda)$, find the MGF of $X$. (Enter $\\lambda$ as $l$) $M(t)=$ [ANS]\nIf $\\lambda=7$, differentiate the MGF you found above to find the first 3 moments of $X$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $X$. $var(X)=$ [ANS]", "answer_v3": [ "e^[l*(e^t-1)]", "7", "56", "497", "7" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0182", "subject": "Probability", "topic": "Random variables", "subtopic": "Generating function", "level": "3", "keywords": [ "moments", "expected value", "expectation", "standard deviation", "variance", "sampling distributions", "binomial distribution", "binomial random variable" ], "problem_v1": "If $Y=\\sum_{i=1}^{25} X_i$ and every $X_i$ is i.i.d with a chi-squared distribution with $14$ degrees of freedom, find the MGF of Y. $M(t)=$ [ANS]\nWhat is the distribution of $Y$? Select all that apply. There may be more than one correct answer. [ANS] A. $exponential(\\lambda=175)$ B. $chi-squared(df=175)$ C. $gamma(\\alpha=1, \\beta=1/175)$ D. $exponential(\\lambda=350)$ E. $gamma(\\alpha=1, \\beta=1/350)$ F. $gamma(\\alpha=2, \\beta=350)$ G. $gamma(\\alpha=175, \\beta=2)$ H. $chi-squared(df=350)$ I. None of the above", "answer_v1": [ "(1-2*t)^{-175}", "GH" ], "answer_type_v1": [ "EX", "MCM" ], "options_v1": [ [], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v2": "If $Y=\\sum_{i=1}^{11} X_i$ and every $X_i$ is i.i.d with a chi-squared distribution with $20$ degrees of freedom, find the MGF of Y. $M(t)=$ [ANS]\nWhat is the distribution of $Y$? Select all that apply. There may be more than one correct answer. [ANS] A. $gamma(\\alpha=1, \\beta=1/110)$ B. $gamma(\\alpha=110, \\beta=2)$ C. $gamma(\\alpha=2, \\beta=220)$ D. $chi-squared(df=110)$ E. $gamma(\\alpha=1, \\beta=1/220)$ F. $exponential(\\lambda=220)$ G. $exponential(\\lambda=110)$ H. $chi-squared(df=220)$ I. None of the above", "answer_v2": [ "(1-2*t)^{-110}", "BH" ], "answer_type_v2": [ "EX", "MCM" ], "options_v2": [ [], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v3": "If $Y=\\sum_{i=1}^{16} X_i$ and every $X_i$ is i.i.d with a chi-squared distribution with $14$ degrees of freedom, find the MGF of Y. $M(t)=$ [ANS]\nWhat is the distribution of $Y$? Select all that apply. There may be more than one correct answer. [ANS] A. $gamma(\\alpha=1, \\beta=1/112)$ B. $exponential(\\lambda=224)$ C. $gamma(\\alpha=112, \\beta=2)$ D. $gamma(\\alpha=2, \\beta=224)$ E. $chi-squared(df=112)$ F. $exponential(\\lambda=112)$ G. $chi-squared(df=224)$ H. $gamma(\\alpha=1, \\beta=1/224)$ I. None of the above", "answer_v3": [ "(1-2*t)^{-112}", "CG" ], "answer_type_v3": [ "EX", "MCM" ], "options_v3": [ [], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ] }, { "id": "Probability_0183", "subject": "Probability", "topic": "Random variables", "subtopic": "Generating function", "level": "3", "keywords": [ "moments", "expected value", "expectation", "standard deviation", "variance", "sampling distributions" ], "problem_v1": "If $Z$ is $gamma(\\alpha,\\beta)$, find the MGF of $Z$. (Enter $\\alpha$ as $a$ and $\\beta$ as $b$) $M(t)=$ [ANS]\nIf $\\alpha=16$ and $\\beta=13$, differentiate the MGF you found above to find the first 3 moments of $Z$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $Z$. $var(Z)=$ [ANS]", "answer_v1": [ "(1-b*t)^{-a}", "208", "45968", "1.07565E+07", "2704" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "If $Z$ is $gamma(\\alpha,\\beta)$, find the MGF of $Z$. (Enter $\\alpha$ as $a$ and $\\beta$ as $b$) $M(t)=$ [ANS]\nIf $\\alpha=3$ and $\\beta=19$, differentiate the MGF you found above to find the first 3 moments of $Z$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $Z$. $var(Z)=$ [ANS]", "answer_v2": [ "(1-b*t)^{-a}", "57", "4332", "411540", "1083" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "If $Z$ is $gamma(\\alpha,\\beta)$, find the MGF of $Z$. (Enter $\\alpha$ as $a$ and $\\beta$ as $b$) $M(t)=$ [ANS]\nIf $\\alpha=7$ and $\\beta=13$, differentiate the MGF you found above to find the first 3 moments of $Z$ about 0. 1st Moment: [ANS]\n2nd Moment: [ANS]\n3rd Moment: [ANS]\nUsing the moments above, calculate the variance of $Z$. $var(Z)=$ [ANS]", "answer_v3": [ "(1-b*t)^{-a}", "91", "9464", "1.10729E+06", "1183" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0184", "subject": "Probability", "topic": "Random variables", "subtopic": "Generating function", "level": "3", "keywords": [ "Random variables", "Moment generating function", "finding the moment generating function of a discrete variable and evaluating it at a given point", "finding the second derivative of the m.g.f. and evaluating it at a given point", "finding the second moment of a discrete variable" ], "problem_v1": "The random variable $X$ has mass function\n$p(x)=\\begin{cases} \\frac{1}{4} & x=1 \\\\ \\\\ \\frac{1}{2} & x=2 \\\\ \\\\ \\frac{1}{4} & x=3 \\end{cases}$\nProvide your answers to two decimal places.\n(a) Find the moment generating function of $X$ and evaluate it at the point $t$=4. [ANS]\n(b) Find the second derivative of the moment generating function and evaluate it at the point $t$=4. [ANS]\n(c) Use the m.g.f. to find the second moment of $X$. [ANS]", "answer_v1": [ "42192.83", "372173.85", "4.50" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The random variable $X$ has mass function\n$p(x)=\\begin{cases} \\frac{1}{4} & x=1 \\\\ \\\\ \\frac{1}{2} & x=2 \\\\ \\\\ \\frac{1}{4} & x=3 \\end{cases}$\nProvide your answers to two decimal places.\n(a) Find the moment generating function of $X$ and evaluate it at the point $t$=1. [ANS]\n(b) Find the second derivative of the moment generating function and evaluate it at the point $t$=1. [ANS]\n(c) Use the m.g.f. to find the second moment of $X$. [ANS]", "answer_v2": [ "9.40", "60.65", "4.50" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The random variable $X$ has mass function\n$p(x)=\\begin{cases} \\frac{1}{4} & x=1 \\\\ \\\\ \\frac{1}{2} & x=2 \\\\ \\\\ \\frac{1}{4} & x=3 \\end{cases}$\nProvide your answers to two decimal places.\n(a) Find the moment generating function of $X$ and evaluate it at the point $t$=2. [ANS]\n(b) Find the second derivative of the moment generating function and evaluate it at the point $t$=2. [ANS]\n(c) Use the m.g.f. to find the second moment of $X$. [ANS]", "answer_v3": [ "130.00", "1018.76", "4.50" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0185", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Bernoulli", "level": "4", "keywords": [ "Expected Value" ], "problem_v1": "$25$ people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then either sits at the table of a friend or at an unoccupied table is none of those present is a friend. Assuming that each of the ${25}\\choose{2}$ pairs of people are, independently, friends with probability $0.6,$ find the expected number of occupied tables. (Hint: One possible approach is to define, for example, $X_3$ to be the random variable whose value is 1 if the third person to arrive sits at an unoccupied table and 0 otherwise.) Answer: The expected number of occupied tables is [ANS]", "answer_v1": [ "1.66666666647902" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$11$ people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then either sits at the table of a friend or at an unoccupied table is none of those present is a friend. Assuming that each of the ${11}\\choose{2}$ pairs of people are, independently, friends with probability $0.8,$ find the expected number of occupied tables. (Hint: One possible approach is to define, for example, $X_3$ to be the random variable whose value is 1 if the third person to arrive sits at an unoccupied table and 0 otherwise.) Answer: The expected number of occupied tables is [ANS]", "answer_v2": [ "1.2499999744" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$16$ people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then either sits at the table of a friend or at an unoccupied table is none of those present is a friend. Assuming that each of the ${16}\\choose{2}$ pairs of people are, independently, friends with probability $0.6,$ find the expected number of occupied tables. (Hint: One possible approach is to define, for example, $X_3$ to be the random variable whose value is 1 if the third person to arrive sits at an unoccupied table and 0 otherwise.) Answer: The expected number of occupied tables is [ANS]", "answer_v3": [ "1.66666595083878" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0186", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "Expected Value" ], "problem_v1": "Consider $n=32$ independent flips of a fair coin. Say that a changeover occurs whenever an outcome differs from the one preceding it. For example, if $n=6$ and the outcome is $T \\ H \\ T \\ T \\ H \\ T,$ then there is a total of 4 changeovers. Find the expected number of changeovers for $n=32$.\nAnswer=[ANS]", "answer_v1": [ "31/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider $n=9$ independent flips of a fair coin. Say that a changeover occurs whenever an outcome differs from the one preceding it. For example, if $n=6$ and the outcome is $T \\ H \\ T \\ T \\ H \\ T,$ then there is a total of 4 changeovers. Find the expected number of changeovers for $n=9$.\nAnswer=[ANS]", "answer_v2": [ "8/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider $n=17$ independent flips of a fair coin. Say that a changeover occurs whenever an outcome differs from the one preceding it. For example, if $n=6$ and the outcome is $T \\ H \\ T \\ T \\ H \\ T,$ then there is a total of 4 changeovers. Find the expected number of changeovers for $n=17$.\nAnswer=[ANS]", "answer_v3": [ "16/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0187", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "1", "keywords": [ "probability", "binomial random variable", "probability dist" ], "problem_v1": "In each part, assume the random variable $X$ has a binomial distribution with the given parameters. Compute the probability of the event.\n(a) $n=6, p=0.4$ $Pr(X=2)=$ [ANS]\n(b) $n=5, p=0.5$ $Pr(X=1)=$ [ANS]\n(c) $n=5, p=0.6$ $Pr(X=3)=$ [ANS]\n(d) $n=5, p=0.3$ $Pr(X=3)=$ [ANS]", "answer_v1": [ "0.31104", "0.15625", "0.3456", "0.1323" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "In each part, assume the random variable $X$ has a binomial distribution with the given parameters. Compute the probability of the event.\n(a) $n=3, p=0.6$ $Pr(X=3)=$ [ANS]\n(b) $n=6, p=0.1$ $Pr(X=2)=$ [ANS]\n(c) $n=3, p=0.6$ $Pr(X=0)=$ [ANS]\n(d) $n=4, p=0.4$ $Pr(X=1)=$ [ANS]", "answer_v2": [ "0.216", "0.098415", "0.064", "0.3456" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "In each part, assume the random variable $X$ has a binomial distribution with the given parameters. Compute the probability of the event.\n(a) $n=4, p=0.8$ $Pr(X=1)=$ [ANS]\n(b) $n=5, p=0.2$ $Pr(X=2)=$ [ANS]\n(c) $n=4, p=0.3$ $Pr(X=4)=$ [ANS]\n(d) $n=5, p=0.3$ $Pr(X=5)=$ [ANS]", "answer_v3": [ "0.0256", "0.2048", "0.0081", "0.00243" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0188", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "Binomial Distribution", "Mean", "Standard Deviation", "probability", "statistics", "binomial" ], "problem_v1": "The Census Bureau reports that 82\\% of Americans over the age of 25 are high school graduates. A survey of randomly selected residents of certain county included 1330 who were over the age of 25, and 1099 of them were high school graduates.\n(a) Find the mean and standard deviation for the number of high school graduates in groups of 1330 Americans over the age of 25.\nMean=[ANS]\nStandard deviation=[ANS]\n(b) Is that county result of 1099 unusually high, or low, or neither? (Enter HIGH or LOW or NEITHER) [ANS]", "answer_v1": [ "1090.6", "14.0109956819635", "NEITHER" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "HIGH", "LOW", "NEITHER" ] ], "problem_v2": "The Census Bureau reports that 82\\% of Americans over the age of 25 are high school graduates. A survey of randomly selected residents of certain county included 1120 who were over the age of 25, and 962 of them were high school graduates.\n(a) Find the mean and standard deviation for the number of high school graduates in groups of 1120 Americans over the age of 25.\nMean=[ANS]\nStandard deviation=[ANS]\n(b) Is that county result of 962 unusually high, or low, or neither? (Enter HIGH or LOW or NEITHER) [ANS]", "answer_v2": [ "918.4", "12.8573714265397", "HIGH" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "HIGH", "LOW", "NEITHER" ] ], "problem_v3": "The Census Bureau reports that 82\\% of Americans over the age of 25 are high school graduates. A survey of randomly selected residents of certain county included 1190 who were over the age of 25, and 986 of them were high school graduates.\n(a) Find the mean and standard deviation for the number of high school graduates in groups of 1190 Americans over the age of 25.\nMean=[ANS]\nStandard deviation=[ANS]\n(b) Is that county result of 986 unusually high, or low, or neither? (Enter HIGH or LOW or NEITHER) [ANS]", "answer_v3": [ "975.8", "13.2530751148554", "NEITHER" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "HIGH", "LOW", "NEITHER" ] ] }, { "id": "Probability_0189", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "Binomial Distribution", "probability" ], "problem_v1": "A quiz consists of 20 multiple-choice questions, each with 5 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 60 \\%. $P(\\mbox{pass})=$ [ANS]", "answer_v1": [ "0.000101729" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A quiz consists of 10 multiple-choice questions, each with 6 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 40 \\%. $P(\\mbox{pass})=$ [ANS]", "answer_v2": [ "0.0697278" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A quiz consists of 10 multiple-choice questions, each with 5 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 50 \\%. $P(\\mbox{pass})=$ [ANS]", "answer_v3": [ "0.0327935" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0190", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "Binomial Distribution", "probability" ], "problem_v1": "The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80 \\% of its flights arriving on time. A test is conducted by randomly selecting $18$ Southwest flights and observing whether they arrive on time. \n(a) Find the probability that at least $11$ flights arrive on time. [ANS]\n(b) Would it be unusual for Southwest to have $4$ flights arrive late? (Enter YES or NO) $\\ $ [ANS]", "answer_v1": [ "0.983719859231655", "NO" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80 \\% of its flights arriving on time. A test is conducted by randomly selecting $10$ Southwest flights and observing whether they arrive on time. \n(a) Find the probability that exactly $9$ flights arrive on time. [ANS]\n(b) Would it be unusual for Southwest to have $3$ flights arrive late? (Enter YES or NO) $\\ $ [ANS]", "answer_v2": [ "0.268435456", "NO" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80 \\% of its flights arriving on time. A test is conducted by randomly selecting $13$ Southwest flights and observing whether they arrive on time. \n(a) Find the probability that exactly $8$ flights arrive late. [ANS]\n(b) Would it be unusual for Southwest to have $4$ flights arrive late? (Enter YES or NO) $\\ $ [ANS]", "answer_v3": [ "0.0010796138496", "NO" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Probability_0191", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability", "binomial distribution" ], "problem_v1": "A man claims to have extrasensory perception (ESP). As a test, a fair coin is flipped $28$ times, and the man is asked to predict the outcome in advance. He gets $23$ out of $28$ correct. What is the probability that he would have done at least this well if he had no ESP? Probability=[ANS]", "answer_v1": [ "0.00045611709356308" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A man claims to have extrasensory perception (ESP). As a test, a fair coin is flipped $20$ times, and the man is asked to predict the outcome in advance. He gets $18$ out of $20$ correct. What is the probability that he would have done at least this well if he had no ESP? Probability=[ANS]", "answer_v2": [ "0.000201225280761719" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A man claims to have extrasensory perception (ESP). As a test, a fair coin is flipped $23$ times, and the man is asked to predict the outcome in advance. He gets $19$ out of $23$ correct. What is the probability that he would have done at least this well if he had no ESP? Probability=[ANS]", "answer_v3": [ "0.00129973888397217" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0192", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [], "problem_v1": "Suppose that 8 dice thrown are thrown and the number, N, of spots showing is noted. Then suppose N coins are tossed, what is the expected number of heads? Expected number of heads=[ANS]", "answer_v1": [ "14" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that 2 dice thrown are thrown and the number, N, of spots showing is noted. Then suppose N coins are tossed, what is the expected number of heads? Expected number of heads=[ANS]", "answer_v2": [ "3.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that 4 dice thrown are thrown and the number, N, of spots showing is noted. Then suppose N coins are tossed, what is the expected number of heads? Expected number of heads=[ANS]", "answer_v3": [ "7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0193", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [], "problem_v1": "Suppose that the number of dice thrown is a binomial random variable with n=22 and p=0.75. What is the expected number of spots showing on the thrown dice? Expected number of spots=[ANS]", "answer_v1": [ "57.75" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the number of dice thrown is a binomial random variable with n=29 and p=0.09. What is the expected number of spots showing on the thrown dice? Expected number of spots=[ANS]", "answer_v2": [ "9.135" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the number of dice thrown is a binomial random variable with n=22 and p=0.32. What is the expected number of spots showing on the thrown dice? Expected number of spots=[ANS]", "answer_v3": [ "24.64" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0194", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability' 'binomial" ], "problem_v1": "Suppose there exists a Hamming code of length 27 that corrects 3 errors. Assuming that the probability of a correct transmission of an individual symbol is 0.86, find the probability that a message transmitted using this code will be correctly received. answer: [ANS]", "answer_v1": [ "0.465449583082719" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose there exists a Hamming code of length 10 that corrects 4 errors. Assuming that the probability of a correct transmission of an individual symbol is 0.73, find the probability that a message transmitted using this code will be correctly received. answer: [ANS]", "answer_v2": [ "0.896316855978706" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose there exists a Hamming code of length 16 that corrects 3 errors. Assuming that the probability of a correct transmission of an individual symbol is 0.77, find the probability that a message transmitted using this code will be correctly received. answer: [ANS]", "answer_v3": [ "0.479650714479689" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0195", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability" ], "problem_v1": "Suppose the number of radios in a household has a binomial distribution with parameters $n=21$ and $p=35$ \\%.\nFind the probability of a household having:\n(a) 10 or 17 radios [ANS]\n(b) 15 or fewer radios [ANS]\n(c) 13 or more radios [ANS]\n(d) fewer than 17 radios [ANS]\n(e) more than 15 radios [ANS]", "answer_v1": [ "0.0851626632079356", "0.999858837183754", "0.010762699840027", "0.999978565965795", "0.000141162816246432" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Suppose the number of TV's in a household has a binomial distribution with parameters $n=8$ and $p=25$ \\%.\nFind the probability of a household having:\n(a) 4 or 6 TV's [ANS]\n(b) 4 or fewer TV's [ANS]\n(c) 4 or more TV's [ANS]\n(d) fewer than 6 TV's [ANS]\n(e) more than 4 TV's [ANS]", "answer_v2": [ "0.090362548828125", "0.972702026367188", "0.113815307617188", "0.995773315429688", "0.0272979736328125" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Suppose the number of cell phones in a household has a binomial distribution with parameters $n=12$ and $p=35$ \\%.\nFind the probability of a household having:\n(a) 5 or 8 cell phones [ANS]\n(b) 6 or fewer cell phones [ANS]\n(c) 6 or more cell phones [ANS]\n(d) fewer than 8 cell phones [ANS]\n(e) more than 6 cell phones [ANS]", "answer_v3": [ "0.223817271447056", "0.915367934878433", "0.21273538272066", "0.974492543001091", "0.0846320651215668" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0196", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "binomial distribution", "statistics", "probability" ], "problem_v1": "Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you replace the card, and draw another card. You repeat this process until you have drawn 19 cards in all. What is the probability of drawing at least 7 clubs? [ANS]\nFor the experiment above, let $X$ denote the number of clubs that are drawn. For this random variable, find its expected value and standard deviation. $E(X)=$ [ANS]\n$\\sigma=$ [ANS]", "answer_v1": [ "0.174876", "4.75", "1.88746" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you replace the card, and draw another card. You repeat this process until you have drawn 15 cards in all. What is the probability of drawing at least 8 diamonds? [ANS]\nFor the experiment above, let $X$ denote the number of diamonds that are drawn. For this random variable, find its expected value and standard deviation. $E(X)=$ [ANS]\n$\\sigma=$ [ANS]", "answer_v2": [ "0.0172998", "3.75", "1.67705" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you replace the card, and draw another card. You repeat this process until you have drawn 16 cards in all. What is the probability of drawing at least 7 hearts? [ANS]\nFor the experiment above, let $X$ denote the number of hearts that are drawn. For this random variable, find its expected value and standard deviation. $E(X)=$ [ANS]\n$\\sigma=$ [ANS]", "answer_v3": [ "0.0795573", "4", "1.73205" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0197", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "binomial distribution", "statistics", "probability" ], "problem_v1": "It is known that a certain basketball player will successfully make a free throw 90.27\\% of the time. Suppose that the basketball player attempts to make 13 free throws. What is the probability that the basketball player will make at least 10 free throws? [ANS]\nLet $X$ be the random variable which denotes the number of free throws that are made by the basketball player. Find the expected value and standard deviation of the random variable. $E(X)=$ [ANS]\n$~ \\sigma=$ [ANS]", "answer_v1": [ "0.96875", "11.7351", "1.06856" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "It is known that a certain lacrosse goalie will successfully make a save 85.58\\% of the time. Suppose that the lacrosse goalie attempts to make 15 saves. What is the probability that the lacrosse goalie will make at least 13 saves? [ANS]\nLet $X$ be the random variable which denotes the number of saves that are made by the lacrosse goalie. Find the expected value and standard deviation of the random variable. $E(X)=$ [ANS]\n$~ \\sigma=$ [ANS]", "answer_v2": [ "0.629606", "12.837", "1.36055" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "It is known that a certain lacrosse goalie will successfully make a save 87.19\\% of the time. Suppose that the lacrosse goalie attempts to make 13 saves. What is the probability that the lacrosse goalie will make at least 11 saves? [ANS]\nLet $X$ be the random variable which denotes the number of saves that are made by the lacrosse goalie. Find the expected value and standard deviation of the random variable. $E(X)=$ [ANS]\n$~ \\sigma=$ [ANS]", "answer_v3": [ "0.77308", "11.3347", "1.20498" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0198", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability", "conditional" ], "problem_v1": "One interpretation of a baseball player's batting average is as the empirical probability of getting a hit each time the player goes to bat. If a player with a batting average of $0.275$ bats $4$ times in a game, and each at-bat is an independent event, what is the probability of the player getting at least on hit in the game? [ANS]", "answer_v1": [ "0.723718" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "One interpretation of a baseball player's batting average is as the empirical probability of getting a hit each time the player goes to bat. If a player with a batting average of $0.208$ bats $5$ times in a game, and each at-bat is an independent event, what is the probability of the player getting at least on hit in the game? [ANS]", "answer_v2": [ "0.68838" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "One interpretation of a baseball player's batting average is as the empirical probability of getting a hit each time the player goes to bat. If a player with a batting average of $0.231$ bats $4$ times in a game, and each at-bat is an independent event, what is the probability of the player getting at least on hit in the game? [ANS]", "answer_v3": [ "0.650292" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0199", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability", "Binomial distribution" ], "problem_v1": "The proportion of adults who own a cell phone in a large Canadian city is believed to be 85\\%. Thirty adults are to be selected at random from this city. Let $X$ be the number in the sample who own a cell phone. Under the assumptions given, the distribution of $X$ is [ANS] A. $Bin(30,25.5)$ B. $N(30,4.5)$ C. $Bin(30,0.85)$ D. $N(25.5,4.5)$", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "The proportion of adults who own a cell phone in a large Canadian city is believed to be 60\\%. Fifty adults are to be selected at random from this city. Let $X$ be the number in the sample who own a cell phone. Under the assumptions given, the distribution of $X$ is [ANS] A. $Bin(50,30)$ B. $N(50,20)$ C. $N(30,20)$ D. $Bin(50,0.6)$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "The proportion of adults who own a cell phone in a large Canadian city is believed to be 70\\%. Forty adults are to be selected at random from this city. Let $X$ be the number in the sample who own a cell phone. Under the assumptions given, the distribution of $X$ is [ANS] A. $Bin(40,28)$ B. $Bin(40,0.7)$ C. $N(28,12)$ D. $N(40,12)$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0200", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "probability", "Binomial distribution", "statistics", "multiple choice", "probability" ], "problem_v1": "Suppose it is believed that the probability a patient will recover from a disease following medication is 0.9. In a group of twenty such patients, the number who recover would have mean and variance respectively given by (to one decimal place):\nMean: [ANS]\nVariance: [ANS]", "answer_v1": [ "18", "1.8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose it is believed that the probability a patient will recover from a disease following medication is 0.7. In a group of fifty such patients, the number who recover would have mean and variance respectively given by (to one decimal place):\nMean: [ANS]\nVariance: [ANS]", "answer_v2": [ "35", "10.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose it is believed that the probability a patient will recover from a disease following medication is 0.7. In a group of twenty such patients, the number who recover would have mean and variance respectively given by (to one decimal place):\nMean: [ANS]\nVariance: [ANS]", "answer_v3": [ "14", "4.2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0201", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "Random variable", "Binomial distribution", "expectation", "variance", "tail probability", "identify Binomial example", "find mean", "variance and a tail probability" ], "problem_v1": "A life insurance salesman operates on the premise that the probability that a man reaching his sixtieth birthday will not live to his sixty-first birthday is $0.05$. On visiting a holiday resort for seniors, he sells $11$ policies to men approaching their sixtieth birthdays. Each policy comes into effect on the birthday of the insured, and pays a fixed sum on death. All $11$ policies can be assumed to be mutually independent. Provide answers to the following to 3 decimal places.\nPart a) What is the expected number of policies that will pay out before the insured parties have reached age 61? [ANS]\nPart b) What is the variance of the number of policies that will pay out before the insured parties have reached age 61? [ANS]\nPart c) What is the probability that at least two policies will pay out before the insured parties have reached age 61? [ANS]", "answer_v1": [ "0.55", "0.5225", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A life insurance salesman operates on the premise that the probability that a man reaching his sixtieth birthday will not live to his sixty-first birthday is $0.06$. On visiting a holiday resort for seniors, he sells $8$ policies to men approaching their sixtieth birthdays. Each policy comes into effect on the birthday of the insured, and pays a fixed sum on death. All $8$ policies can be assumed to be mutually independent. Provide answers to the following to 3 decimal places.\nPart a) What is the expected number of policies that will pay out before the insured parties have reached age 61? [ANS]\nPart b) What is the variance of the number of policies that will pay out before the insured parties have reached age 61? [ANS]\nPart c) What is the probability that at least two policies will pay out before the insured parties have reached age 61? [ANS]", "answer_v2": [ "0.48", "0.4512", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A life insurance salesman operates on the premise that the probability that a man reaching his sixtieth birthday will not live to his sixty-first birthday is $0.05$. On visiting a holiday resort for seniors, he sells $9$ policies to men approaching their sixtieth birthdays. Each policy comes into effect on the birthday of the insured, and pays a fixed sum on death. All $9$ policies can be assumed to be mutually independent. Provide answers to the following to 3 decimal places.\nPart a) What is the expected number of policies that will pay out before the insured parties have reached age 61? [ANS]\nPart b) What is the variance of the number of policies that will pay out before the insured parties have reached age 61? [ANS]\nPart c) What is the probability that at least two policies will pay out before the insured parties have reached age 61? [ANS]", "answer_v3": [ "0.45", "0.4275", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0202", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "3", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "70\\% of the employees in a specialized department of a large software firm are computer science graduates. A project team is made up of 10 employees. Part a) What is the probability to 3 decimal digits that all the project team members are computer science graduates? [ANS]\nPart b) What is the probability to 3 decimal digits that exactly 3 of the project team members are computer science graduates? [ANS]\nPart c) What is the most likely number of computer science graduates among the 10 project team members? Your answer should be an integer. If there are two possible answers, please select the smaller of the two integers. [ANS]\nPart d) There are 53 such projects running at the same time and each project team consists of 10 employees as described. On how many of the 53 project teams do you expect there to be exactly 3 computer science graduates? Give your answer to 1 decimal place. [ANS]\nPart e) I meet 40 employees at random. What is the probability that the second employee I meet is the first one who is a computer science graduate? Give your answer to 3 decimal places. [ANS]\nPart f) I meet 46 employees at random on a daily basis. What is the mean number of computer science graduates among the 46 that I meet? Give your answer to one decimal place. [ANS]", "answer_v1": [ "0.0282", "0.009", "7", "0.48", "0.21", "32.2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "80\\% of the employees in a specialized department of a large software firm are computer science graduates. A project team is made up of 8 employees. Part a) What is the probability to 3 decimal digits that all the project team members are computer science graduates? [ANS]\nPart b) What is the probability to 3 decimal digits that exactly 3 of the project team members are computer science graduates? [ANS]\nPart c) What is the most likely number of computer science graduates among the 8 project team members? Your answer should be an integer. If there are two possible answers, please select the smaller of the two integers. [ANS]\nPart d) There are 43 such projects running at the same time and each project team consists of 8 employees as described. On how many of the 43 project teams do you expect there to be exactly 3 computer science graduates? Give your answer to 1 decimal place. [ANS]\nPart e) I meet 30 employees at random. What is the probability that the fourth employee I meet is the first one who is a computer science graduate? Give your answer to 3 decimal places. [ANS]\nPart f) I meet 45 employees at random on a daily basis. What is the mean number of computer science graduates among the 45 that I meet? Give your answer to one decimal place. [ANS]", "answer_v2": [ "0.1678", "0.0092", "7", "0.4", "0.0064", "36" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "75\\% of the employees in a specialized department of a large software firm are computer science graduates. A project team is made up of 8 employees. Part a) What is the probability to 3 decimal digits that all the project team members are computer science graduates? [ANS]\nPart b) What is the probability to 3 decimal digits that exactly 3 of the project team members are computer science graduates? [ANS]\nPart c) What is the most likely number of computer science graduates among the 8 project team members? Your answer should be an integer. If there are two possible answers, please select the smaller of the two integers. [ANS]\nPart d) There are 45 such projects running at the same time and each project team consists of 8 employees as described. On how many of the 45 project teams do you expect there to be exactly 3 computer science graduates? Give your answer to 1 decimal place. [ANS]\nPart e) I meet 40 employees at random. What is the probability that the first employee I meet is the first one who is a computer science graduate? Give your answer to 3 decimal places. [ANS]\nPart f) I meet 48 employees at random on a daily basis. What is the mean number of computer science graduates among the 48 that I meet? Give your answer to one decimal place. [ANS]", "answer_v3": [ "0.1001", "0.0231", "6", "1.04", "0.75", "36" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0203", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability", "random variables" ], "problem_v1": "A sign on the pumps at a gas station encourages customers to have their oil checked, and claims that one out of 7 cars needs to have oil added. If this is true, what is the probability of each of the following: A. One out of the next four cars needs oil. Probability=[ANS]\nB. Two out of the next eight cars needs oil. Probability=[ANS]\nC. 10 out of the next 40 cars needs oil. Probability=[ANS]", "answer_v1": [ "0.359850062473969", "0.226611118059409", "0.0294332264175672" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A sign on the pumps at a gas station encourages customers to have their oil checked, and claims that one out of 3 cars needs to have oil added. If this is true, what is the probability of each of the following: A. One out of the next four cars needs oil. Probability=[ANS]\nB. Two out of the next eight cars needs oil. Probability=[ANS]\nC. 10 out of the next 40 cars needs oil. Probability=[ANS]", "answer_v2": [ "0.395061728395062", "0.273129096174364", "0.0748637610183203" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A sign on the pumps at a gas station encourages customers to have their oil checked, and claims that one out of 4 cars needs to have oil added. If this is true, what is the probability of each of the following: A. One out of the next four cars needs oil. Probability=[ANS]\nB. Two out of the next eight cars needs oil. Probability=[ANS]\nC. 10 out of the next 40 cars needs oil. Probability=[ANS]", "answer_v3": [ "0.421875", "0.31146240234375", "0.144364346356257" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0204", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability", "random variables" ], "problem_v1": "In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 12\\% of voters are Independent. A survey asked 32 people to identify themselves as Democrat, Republican, or Independent. A. What is the probability that none of the people are Independent? Probability=[ANS]\nB. What is the probability that fewer than 6 are Independent? Probability=[ANS]\nC. What is the probability that more than 3 people are Independent? Probability=[ANS]", "answer_v1": [ "0.0167280573527069", "0.821230494376496", "0.545602619393601" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 8\\% of voters are Independent. A survey asked 39 people to identify themselves as Democrat, Republican, or Independent. A. What is the probability that none of the people are Independent? Probability=[ANS]\nB. What is the probability that fewer than 4 are Independent? Probability=[ANS]\nC. What is the probability that more than 2 people are Independent? Probability=[ANS]", "answer_v2": [ "0.0387012744244995", "0.619350805212259", "0.613206733681799" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 9\\% of voters are Independent. A survey asked 32 people to identify themselves as Democrat, Republican, or Independent. A. What is the probability that none of the people are Independent? Probability=[ANS]\nB. What is the probability that fewer than 5 are Independent? Probability=[ANS]\nC. What is the probability that more than 3 people are Independent? Probability=[ANS]", "answer_v3": [ "0.0489017672496492", "0.843810628523103", "0.324436904822212" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0205", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Binomial", "level": "2", "keywords": [ "probability", "random variables" ], "problem_v1": "In the game of blackjack as played in casinos in Las Vegas, Atlantic City, Niagara Falls, as well as many other cities, the dealer has the advantage. Most players do not play very well. As a result, the probability that the average player wins a hand is about 0.44. Find the probability that an average player wins A. twice in 5 hands. Probability=[ANS]\nB. 11 or more times in 25 hands. Probability=[ANS]\nThere are several books that teach blackjack players the \"basic strategy\" which increases the probability of winning any hand to 0.55. Assuming that the player plays the basic strategy, find the probability that he or she wins C. twice in 5 hands. Probability=[ANS]\nD. 11 or more times in 25 hands. Probability=[ANS]", "answer_v1": [ "0.339992576", "0.576473944285932", "0.275653125", "0.904017028046784" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "In the game of blackjack as played in casinos in Las Vegas, Atlantic City, Niagara Falls, as well as many other cities, the dealer has the advantage. Most players do not play very well. As a result, the probability that the average player wins a hand is about 0.3. Find the probability that an average player wins A. twice in 5 hands. Probability=[ANS]\nB. 10 or more times in 24 hands. Probability=[ANS]\nThere are several books that teach blackjack players the \"basic strategy\" which increases the probability of winning any hand to 0.46. Assuming that the player plays the basic strategy, find the probability that he or she wins C. twice in 5 hands. Probability=[ANS]\nD. 10 or more times in 24 hands. Probability=[ANS]", "answer_v2": [ "0.3087", "0.15278161855166", "0.333193824", "0.734299473474194" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "In the game of blackjack as played in casinos in Las Vegas, Atlantic City, Niagara Falls, as well as many other cities, the dealer has the advantage. Most players do not play very well. As a result, the probability that the average player wins a hand is about 0.35. Find the probability that an average player wins A. twice in 5 hands. Probability=[ANS]\nB. 10 or more times in 24 hands. Probability=[ANS]\nThere are several books that teach blackjack players the \"basic strategy\" which increases the probability of winning any hand to 0.47. Assuming that the player plays the basic strategy, find the probability that he or she wins C. twice in 5 hands. Probability=[ANS]\nD. 10 or more times in 24 hands. Probability=[ANS]", "answer_v3": [ "0.336415625", "0.31335011400056", "0.328869293", "0.765667267290072" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0206", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "2", "keywords": [ "Probability", "Central Limit Theorem" ], "problem_v1": "An airline company is considering a new policy of booking as many as 338 persons on an airplane that can seat only 310. (Past studies have revealed that only 86\\% of the booked passengers actually arrive for the flight.) Estimate the probability that if the company books 338 persons. not enough seats will be available. [ANS]", "answer_v1": [ "0.000945191345615911" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An airline company is considering a new policy of booking as many as 170 persons on an airplane that can seat only 170. (Past studies have revealed that only 90\\% of the booked passengers actually arrive for the flight.) Estimate the probability that if the company books 170 persons. not enough seats will be available. [ANS]", "answer_v2": [ "3.83817527056307E-06" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An airline company is considering a new policy of booking as many as 228 persons on an airplane that can seat only 210. (Past studies have revealed that only 86\\% of the booked passengers actually arrive for the flight.) Estimate the probability that if the company books 228 persons. not enough seats will be available. [ANS]", "answer_v3": [ "0.00295955317861661" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0207", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "2", "keywords": [ "Probability", "Central Limit Theorem" ], "problem_v1": "A multiple-choice test consists of 28 questions with possible answers of a, b, c, d, e. Estimate the probability that with random guessing, the number of correct answers is at least 12. [ANS]", "answer_v1": [ "0.00265592018484538" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A multiple-choice test consists of 20 questions with possible answers of a, b, c, d, e, f. Estimate the probability that with random guessing, the number of correct answers is at least 7. [ANS]", "answer_v2": [ "0.0287165588294146" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A multiple-choice test consists of 23 questions with possible answers of a, b, c, d, e. Estimate the probability that with random guessing, the number of correct answers is at least 9. [ANS]", "answer_v3": [ "0.0210254756710387" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0208", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "2", "keywords": [ "Probability", "Central Limit Theorem", "Normal", "Approximate" ], "problem_v1": "Use normal approximation to estimate the probability of getting at most 53 girls in 100 births. Assume that boys and girls are equally likely. [ANS]", "answer_v1": [ "0.758036346793126" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use normal approximation to estimate the probability of getting less than 45 girls in 100 births. Assume that boys and girls are equally likely. [ANS]", "answer_v2": [ "0.135666059959802" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use normal approximation to estimate the probability of getting more than 48 girls in 100 births. Assume that boys and girls are equally likely. [ANS]", "answer_v3": [ "0.61791142120275" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0209", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "3", "keywords": [], "problem_v1": "The probability that a skeet shooter can hit 30 or more targets in 35 total shots is 0.000119822427984251. Find the probability of this shooter hitting the target when he shoots once at a single target. Probability of hitting target with one shot=[ANS]", "answer_v1": [ "0.55" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The probability that a skeet shooter can hit 13 or more targets in 15 total shots is 7.94801681011092e-05. Find the probability of this shooter hitting the target when he shoots once at a single target. Probability of hitting target with one shot=[ANS]", "answer_v2": [ "0.36" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The probability that a skeet shooter can hit 15 or more targets in 20 total shots is 0.00216685057462473. Find the probability of this shooter hitting the target when he shoots once at a single target. Probability of hitting target with one shot=[ANS]", "answer_v3": [ "0.41" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0210", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "3", "keywords": [], "problem_v1": "Under the assumption that a pair of dice is fair, the probability is 0.95 that the number of 8's appearing in 324 throws of the dice will lie within $45 \\pm K$. \u00a0 What is $K$? $K=$ [ANS]", "answer_v1": [ "12.2006774123215" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Under the assumption that a pair of dice is fair, the probability is 0.81 that the number of 3's appearing in 360 throws of the dice will lie within $20 \\pm K$. \u00a0 What is $K$? $K=$ [ANS]", "answer_v2": [ "5.69595366683493" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Under the assumption that a pair of dice is fair, the probability is 0.86 that the number of 5's appearing in 324 throws of the dice will lie within $36 \\pm K$. \u00a0 What is $K$? $K=$ [ANS]", "answer_v3": [ "8.34833474912602" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0211", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "2", "keywords": [ "probability", "binomial distribution" ], "problem_v1": "The standard deviation of the sampling distribution of the sample mean is also called the: [ANS] A. standard error of the mean B. population standard deviation C. finite population correction factor D. central limit theorem\nIf a random sample of size $n$ is drawn from a normal population, then the sampling distribution of the sample mean $\\bar{X}$ will be: [ANS] A. approximately normal only for $n > 30$ B. normal only for $n > 30$ C. normal for all values of $n$ D. approximately normal for all values of $n$", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "The standard deviation of the sampling distribution of the sample mean is also called the: [ANS] A. finite population correction factor B. population standard deviation C. central limit theorem D. standard error of the mean\nGiven that $X$ is a binomial random variable, the binomial probability $P(X \\leq x)$ is approximated by the area under a normal curve to the left of [ANS] A. $x+0.5$ B. $x-5$ C. $x$ D. $-x$", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The standard deviation of the sampling distribution of the sample mean is also called the: [ANS] A. finite population correction factor B. standard error of the mean C. population standard deviation D. central limit theorem\nSuppose that $X$ is a binomial random variable with n=20 and p=0.55. Using normal approximation with continuity correction, what is (approximately) $P(X \\geq 12)$? [ANS] A. 0.4111 B. 0.3265 C. 0.4143 D. 0.5000", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0212", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "3", "keywords": [ "probability", "binomial distribution" ], "problem_v1": "Given that $X$ is a binomial random variable, the binomial probability $P(X=x)$ is approximated by the area under a normal curve between [ANS] A. $x-0.5$ and $x+0.5$ B. $1-x$ and $1+x$ C. $0.0$ and $x+0.5$ D. $x-0.5$ and $0.0$\nThe central limit theorem states that, if a random sample of size $n$ is drawn from a population, then the sampling distribution of the sample mean $\\bar{X}$: [ANS] A. has the same variance as the population B. is approximately normal if $n < 30$ C. is approximately normal if $n > 30$ D. is approximately normal if the underlying population is normal", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Given that $X$ is a binomial random variable, the binomial probability $P(X=x)$ is approximated by the area under a normal curve between [ANS] A. $0.0$ and $x+0.5$ B. $1-x$ and $1+x$ C. $x-0.5$ and $0.0$ D. $x-0.5$ and $x+0.5$\nIf two random samples of sizes $n_1$ and $n_2$ are selected independently from two non-normally distributed populations, then the sampling distribution of the sample mean difference $\\bar{X_1}-\\bar{X_2}$, is [ANS] A. approximately normal only if $n_1$ and $n_2$ are both larger than 30 B. approximately normal regardless of $n_1$ and $n_2$ C. always non-normal D. always normal", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Given that $X$ is a binomial random variable, the binomial probability $P(X=x)$ is approximated by the area under a normal curve between [ANS] A. $0.0$ and $x+0.5$ B. $x-0.5$ and $x+0.5$ C. $1-x$ and $1+x$ D. $x-0.5$ and $0.0$\nIf all possible samples of size $n$ are drawn from an infinite population with a mean of $\\mu$ and a standard deviation of $\\sigma$, then the standard error of the sample mean is inversely proportional to [ANS] A. $\\sqrt{n}$ B. $\\sigma$ C. $n$ D. $\\mu$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0213", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Normal approximation to binomial", "level": "3", "keywords": [ "probability", "normal distribution", "finite population", "sample size" ], "problem_v1": "If the standard error of the sampling distribution of the sample proportion is 0.0229 for samples of size 400, then the population proportion must be either [ANS] A. 0.3 or 0.7 B. 0.2 or 0.8 C. 0.5 or 0.5 D. 0.4 or 0.6\nGiven that $X$ is a binomial random variable, the binomial probability $P(X \\geq x)$ is approximated by the area under a normal curve to the right of [ANS] A. $x+1$ B. $x+0.5$ C. $x-0.5$ D. $x-1$", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "If the standard error of the sampling distribution of the sample proportion is 0.0229 for samples of size 400, then the population proportion must be either [ANS] A. 0.5 or 0.5 B. 0.2 or 0.8 C. 0.4 or 0.6 D. 0.3 or 0.7\nAs a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if [ANS] A. $np$ and $n(1-p)$ are both greater than 5 B. the underlying population proportion is normal C. the sample size $n$ is greater than 30 D. the population proportion $p$ is close to 0.50", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "If the standard error of the sampling distribution of the sample proportion is 0.0229 for samples of size 400, then the population proportion must be either [ANS] A. 0.5 or 0.5 B. 0.3 or 0.7 C. 0.2 or 0.8 D. 0.4 or 0.6\nSuppose that $X$ is a binomial random variable with n=16 and p=0.35. Using normal approximation with continuity correction, what is (approximately) $P(X=7)$? [ANS] A. 0.1589 B. 0.1854 C. 0 D. 0.1740", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0214", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Geometric", "level": "4", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "The life times, $Y$ in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.7 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if the life time exceeds 1 year, or as a failure otherwise.\nPart a) Find the probability to 3 decimal places that the first success occurs in the fifth observation. [ANS]\nPart b) Find the probability to 3 decimal places of the second success occurring in the 8th observation given that the first success occurred in the 3rd observation. [ANS]\nPart c) Find the probability to 2 decimal places that the first success occurs in an odd-numbered observation. That is, the first success occurs in the 1st or 3rd or 5th or 7th (and so on) observation. [ANS]", "answer_v1": [ "0.0801", "0.0801", "0.5681" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The life times, $Y$ in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.5 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if the life time exceeds 1 year, or as a failure otherwise.\nPart a) Find the probability to 3 decimal places that the first success occurs in the fifth observation. [ANS]\nPart b) Find the probability to 3 decimal places of the second success occurring in the 8th observation given that the first success occurred in the 3rd observation. [ANS]\nPart c) Find the probability to 2 decimal places that the first success occurs in an odd-numbered observation. That is, the first success occurs in the 1st or 3rd or 5th or 7th (and so on) observation. [ANS]", "answer_v2": [ "0.0756", "0.0756", "0.5363" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The life times, $Y$ in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.55 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if the life time exceeds 1 year, or as a failure otherwise.\nPart a) Find the probability to 3 decimal places that the first success occurs in the fifth observation. [ANS]\nPart b) Find the probability to 3 decimal places of the second success occurring in the 8th observation given that the first success occurred in the 3rd observation. [ANS]\nPart c) Find the probability to 2 decimal places that the first success occurs in an odd-numbered observation. That is, the first success occurs in the 1st or 3rd or 5th or 7th (and so on) observation. [ANS]", "answer_v3": [ "0.0799", "0.0799", "0.5442" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0216", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Negative binomial", "level": "3", "keywords": [ "Random variables", "Geometric distribution", "Negative Binomial distribution", "identify Geometric distribution example", "find probabilities for a Geometric distribution", "identify Negative Binomial distribution example", "find a probability for a Negative Binomial variable" ], "problem_v1": "A telephone saleswoman arranges a sequence of interviews of potential customers in order to sell them an insurance policy. She believes that her success rate in completing a sale in any interview is $11$ \\%. Provide answers to the following to 3 decimal places.\nPart a) What is the probability that she fails to make a sale on the first five interviews? [ANS]\nPart b) What is the probability that she makes her first sale on the fourth interview? [ANS]\nPart c) What is the probability that the second sale is made on the eighth interview? [ANS]", "answer_v1": [ "0.558", "0.078", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A telephone saleswoman arranges a sequence of interviews of potential customers in order to sell them an insurance policy. She believes that her success rate in completing a sale in any interview is $8$ \\%. Provide answers to the following to 3 decimal places.\nPart a) What is the probability that she fails to make a sale on the first five interviews? [ANS]\nPart b) What is the probability that she makes her first sale on the fourth interview? [ANS]\nPart c) What is the probability that the second sale is made on the eighth interview? [ANS]", "answer_v2": [ "0.659", "0.062", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A telephone saleswoman arranges a sequence of interviews of potential customers in order to sell them an insurance policy. She believes that her success rate in completing a sale in any interview is $9$ \\%. Provide answers to the following to 3 decimal places.\nPart a) What is the probability that she fails to make a sale on the first five interviews? [ANS]\nPart b) What is the probability that she makes her first sale on the fourth interview? [ANS]\nPart c) What is the probability that the second sale is made on the eighth interview? [ANS]", "answer_v3": [ "0.624", "0.068", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0217", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "3", "keywords": [ "Joint Distribution" ], "problem_v1": "Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is $2.2$. What is the probability that there will be\n(a) at least $4$ such accidents in the next month? [ANS]\n(b) at most $5$ such accidents in the next $2$ months? [ANS]\n(c) exactly $4$ such accidents in the next $5$ months? [ANS]", "answer_v1": [ "0.180647578296662", "0.719911520009197", "0.0101887333862495" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is $2.2$. What is the probability that there will be\n(a) more than $2$ such accidents in the next month? [ANS]\n(b) at least $3$ such accidents in the next $3$ months? [ANS]\n(c) exactly $4$ such accidents in the next $4$ months? [ANS]", "answer_v2": [ "0.377286250003684", "0.960032387056843", "0.0376641357222037" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is $2.2$. What is the probability that there will be\n(a) less than $3$ such accidents in the next month? [ANS]\n(b) at most $4$ such accidents in the next $2$ months? [ANS]\n(c) exactly $5$ such accidents in the next $5$ months? [ANS]", "answer_v3": [ "0.622713749996316", "0.551183808544316", "0.0224152134497488" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0218", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "2", "keywords": [ "Poisson Distribution" ], "problem_v1": "A statistics professor finds that when she schedules an office hour for student help, an average of $2.9$ students arrive. Find the probability that in a randomly selected office hour, the number of student arrivals is $6$. [ANS]", "answer_v1": [ "0.0454570756973534" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A math professor finds that when she schedules an office hour for student help, an average of $1.5$ students arrive. Find the probability that in a randomly selected office hour, the number of student arrivals is $3$. [ANS]", "answer_v2": [ "0.125510715115285" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A math professor finds that when she schedules an office hour for student help, an average of $1.9$ students arrive. Find the probability that in a randomly selected office hour, the number of student arrivals is $4$. [ANS]", "answer_v3": [ "0.08121638346653" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0219", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "2", "keywords": [ "Poisson Distribution" ], "problem_v1": "Given that $x$ is a random variable having a Poisson distribution, compute the following:\n(a) $P(x=7)$ when $\\mu=3.5$ $P(x)=$ [ANS]\n(b) $P(x \\leq 6)$ when $\\mu=4.5$ $P(x)=$ [ANS]\n(c) $P(x > 3)$ when $\\mu=2$ $P(x)=$ [ANS]\n(d) $P(x < 6)$ when $\\mu=3.5$ $P(x)=$ [ANS]", "answer_v1": [ "0.038549174937634", "0.831050578725412", "0.142876539501453", "0.857613553095779" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Given that $x$ is a random variable having a Poisson distribution, compute the following:\n(a) $P(x=1)$ when $\\mu=6$ $P(x)=$ [ANS]\n(b) $P(x \\leq 2)$ when $\\mu=2.5$ $P(x)=$ [ANS]\n(c) $P(x > 9)$ when $\\mu=2$ $P(x)=$ [ANS]\n(d) $P(x < 2)$ when $\\mu=2$ $P(x)=$ [ANS]", "answer_v2": [ "0.0148725130599982", "0.54381311588333", "4.64980750171096E-05", "0.406005849709838" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Given that $x$ is a random variable having a Poisson distribution, compute the following:\n(a) $P(x=3)$ when $\\mu=4$ $P(x)=$ [ANS]\n(b) $P(x \\leq 3)$ when $\\mu=3.5$ $P(x)=$ [ANS]\n(c) $P(x > 2)$ when $\\mu=2.5$ $P(x)=$ [ANS]\n(d) $P(x < 8)$ when $\\mu=5.5$ $P(x)=$ [ANS]", "answer_v3": [ "0.195366814813165", "0.536632667900785", "0.45618688411667", "0.809485282519508" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0220", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "3", "keywords": [ "Poisson Distribution" ], "problem_v1": "The mean number of patients admitted per day to the emergency room of a small hospital is $2.5$. If, on any given day, there are only $6$ beds available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients? answer: [ANS]", "answer_v1": [ "0.0141873119909133" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The mean number of patients admitted per day to the emergency room of a small hospital is $0.5$. If, on any given day, there are only $9$ beds available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients? answer: [ANS]", "answer_v2": [ "1.70967018320312E-10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The mean number of patients admitted per day to the emergency room of a small hospital is $1$. If, on any given day, there are only $6$ beds available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients? answer: [ANS]", "answer_v3": [ "8.32411492880381E-05" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0221", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "3", "keywords": [ "Poisson Distribution" ], "problem_v1": "A certain typing agency employs two typists. The average number of errors per article is $4$ when typed by the first typist and $3.3$ when typed by the second. If your article is equally likely to be typed by either typist, find the probability that it will have no errors. [ANS]", "answer_v1": [ "0.0275994031449871" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A certain typing agency employs two typists. The average number of errors per article is $1.3$ when typed by the first typist and $4.8$ when typed by the second. If your article is equally likely to be typed by either typist, find the probability that it will have no errors. [ANS]", "answer_v2": [ "0.140380770041516" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A certain typing agency employs two typists. The average number of errors per article is $2.2$ when typed by the first typist and $3.4$ when typed by the second. If your article is equally likely to be typed by either typist, find the probability that it will have no errors. [ANS]", "answer_v3": [ "0.07208821416133" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0222", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "3", "keywords": [ "statistics", "continuous random variables", "probability", "uniform distribution" ], "problem_v1": "The number of major faults on a randomly chosen 1 km stretch of highway has a Poisson distribution with mean 1.7. The random variable $X$ is the distance (in km) between two successive major faults on the highway. Part a) What is the probability of having at least one major fault in the next 2 km stretch on the highway? Give your answer to 3 decimal places. [ANS]\nPart b) Which of the following describes the distribution of $X$, the distance between two successive major faults on the highway? [ANS] A. $X \\sim \\mbox{Exponential}(\\mbox{mean}=\\frac{1}{1.7})$ B. $X \\sim \\mbox{Poisson}(1.7)$ C. $X \\sim \\mbox{Exponential}(\\mbox{mean}=\\frac{1}{2 \\cdot 1.7})$ D. $X \\sim \\mbox{Poisson}(2 \\cdot 1.7)$ E. $X \\sim \\mbox{Exponential}(\\mbox{mean}=2 \\cdot 1.7)$\nPart c) What is the mean distance (in km) and standard deviation between successive major faults? [ANS] A. mean=0.5882; standard deviation=0.3460 B. mean=3.4000; standard deviation=3.4000 C. mean=0.5882; standard deviation=0.5882 D. mean=1.7; standard deviation=1.7 E. mean=0.2941; standard deviation=0.2941\nPart d) What is the median distance (in km) between successive major faults? Give your answer to 2 decimal places. [ANS]\nPart e) What is the probability you must travel more than 3 km before encountering the next four major faults? Give your answer to 3 decimal places. [ANS]\nPart f) By expressing the problem as a sum of independent Exponential random variables and applying the Central Limit Theorem, find the approximate probability that you must travel more than 25 km before encountering the next 33 major faults? Give your answer to 3 decimal places. Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps. [ANS]", "answer_v1": [ "0.9666", "A", "C", "0.41", "0.251", "0.049" ], "answer_type_v1": [ "NV", "MCS", "MCS", "NV", "NV", "NV" ], "options_v1": [ [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [], [], [] ], "problem_v2": "The number of major faults on a randomly chosen 1 km stretch of highway has a Poisson distribution with mean 1.2. The random variable $X$ is the distance (in km) between two successive major faults on the highway. Part a) What is the probability of having at least one major fault in the next 2 km stretch on the highway? Give your answer to 3 decimal places. [ANS]\nPart b) Which of the following describes the distribution of $X$, the distance between two successive major faults on the highway? [ANS] A. $X \\sim \\mbox{Poisson}(1.2)$ B. $X \\sim \\mbox{Exponential}(\\mbox{mean}=\\frac{1}{2 \\cdot 1.2})$ C. $X \\sim \\mbox{Poisson}(2 \\cdot 1.2)$ D. $X \\sim \\mbox{Exponential}(\\mbox{mean}=\\frac{1}{1.2})$ E. $X \\sim \\mbox{Exponential}(\\mbox{mean}=2 \\cdot 1.2)$\nPart c) What is the mean distance (in km) and standard deviation between successive major faults? [ANS] A. mean=2.4000; standard deviation=2.4000 B. mean=0.4167; standard deviation=0.4167 C. mean=1.2; standard deviation=1.2 D. mean=0.8333; standard deviation=0.8333 E. mean=0.8333; standard deviation=0.6944\nPart d) What is the median distance (in km) between successive major faults? Give your answer to 2 decimal places. [ANS]\nPart e) What is the probability you must travel more than 3 km before encountering the next four major faults? Give your answer to 3 decimal places. [ANS]\nPart f) By expressing the problem as a sum of independent Exponential random variables and applying the Central Limit Theorem, find the approximate probability that you must travel more than 25 km before encountering the next 33 major faults? Give your answer to 3 decimal places. Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps. [ANS]", "answer_v2": [ "0.9093", "D", "D", "0.58", "0.515", "0.699" ], "answer_type_v2": [ "NV", "MCS", "MCS", "NV", "NV", "NV" ], "options_v2": [ [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [], [], [] ], "problem_v3": "The number of major faults on a randomly chosen 1 km stretch of highway has a Poisson distribution with mean 1.4. The random variable $X$ is the distance (in km) between two successive major faults on the highway. Part a) What is the probability of having at least one major fault in the next 2 km stretch on the highway? Give your answer to 3 decimal places. [ANS]\nPart b) Which of the following describes the distribution of $X$, the distance between two successive major faults on the highway? [ANS] A. $X \\sim \\mbox{Exponential}(\\mbox{mean}=2 \\cdot 1.4)$ B. $X \\sim \\mbox{Poisson}(1.4)$ C. $X \\sim \\mbox{Poisson}(2 \\cdot 1.4)$ D. $X \\sim \\mbox{Exponential}(\\mbox{mean}=\\frac{1}{2 \\cdot 1.4})$ E. $X \\sim \\mbox{Exponential}(\\mbox{mean}=\\frac{1}{1.4})$\nPart c) What is the mean distance (in km) and standard deviation between successive major faults? [ANS] A. mean=2.8000; standard deviation=2.8000 B. mean=0.7143; standard deviation=0.7143 C. mean=0.7143; standard deviation=0.5102 D. mean=1.4; standard deviation=1.4 E. mean=0.3571; standard deviation=0.3571\nPart d) What is the median distance (in km) between successive major faults? Give your answer to 2 decimal places. [ANS]\nPart e) What is the probability you must travel more than 3 km before encountering the next four major faults? Give your answer to 3 decimal places. [ANS]\nPart f) By expressing the problem as a sum of independent Exponential random variables and applying the Central Limit Theorem, find the approximate probability that you must travel more than 25 km before encountering the next 33 major faults? Give your answer to 3 decimal places. Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps. [ANS]", "answer_v3": [ "0.9392", "E", "B", "0.5", "0.395", "0.364" ], "answer_type_v3": [ "NV", "MCS", "MCS", "NV", "NV", "NV" ], "options_v3": [ [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [], [], [] ] }, { "id": "Probability_0223", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "4", "keywords": [ "probability", "Poisson" ], "problem_v1": "Complaints about an Internet brokerage firm occur at a rate of 6 per day. The number of complaints appears to be Poisson distributed. A. Find the probability that the firm receives 7 or more complaints in a day. Probability=[ANS]\nB. Find the probability that the firm receives 28 or more complaints in a 5-day period. Probability=[ANS]", "answer_v1": [ "0.393697217587409", "0.667130915954477" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Complaints about an Internet brokerage firm occur at a rate of 3 per day. The number of complaints appears to be Poisson distributed. A. Find the probability that the firm receives 5 or more complaints in a day. Probability=[ANS]\nB. Find the probability that the firm receives 8 or more complaints in a 3-day period. Probability=[ANS]", "answer_v2": [ "0.184736755476228", "0.676103035687104" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Complaints about an Internet brokerage firm occur at a rate of 4 per day. The number of complaints appears to be Poisson distributed. A. Find the probability that the firm receives 5 or more complaints in a day. Probability=[ANS]\nB. Find the probability that the firm receives 9 or more complaints in a 3-day period. Probability=[ANS]", "answer_v3": [ "0.371163064820127", "0.844972218232537" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0224", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "2", "keywords": [ "probability", "random variables" ], "problem_v1": "Flaws in a carpet tend to occur randomly and independently at a rate of one every 270 square feet. What is the probability that a carpet that is 8 feet by 13 feet contains no flaws? Probability=[ANS]", "answer_v1": [ "0.680324638494327" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Flaws in a carpet tend to occur randomly and independently at a rate of one every 160 square feet. What is the probability that a carpet that is 9 feet by 10 feet contains no flaws? Probability=[ANS]", "answer_v2": [ "0.569782824730923" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Flaws in a carpet tend to occur randomly and independently at a rate of one every 200 square feet. What is the probability that a carpet that is 8 feet by 11 feet contains no flaws? Probability=[ANS]", "answer_v3": [ "0.644036421083141" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0225", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "3", "keywords": [ "probability", "Poisson" ], "problem_v1": "The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 4.3 per week. Find the probability of the following events. A. No accidents occur in one week. Probability=[ANS]\nB. 7 or more accidents occur in a week. Probability=[ANS]\nC. One accident occurs today. Probability=[ANS]", "answer_v1": [ "0.0135685590122009", "0.144210015361258", "0.332345283273427" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 3.3 per week. Find the probability of the following events. A. No accidents occur in one week. Probability=[ANS]\nB. 10 or more accidents occur in a week. Probability=[ANS]\nC. One accident occurs today. Probability=[ANS]", "answer_v2": [ "0.03688316740124", "0.00219454331273772", "0.294223307095113" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 3.5 per week. Find the probability of the following events. A. No accidents occur in one week. Probability=[ANS]\nB. 7 or more accidents occur in a week. Probability=[ANS]\nC. One accident occurs today. Probability=[ANS]", "answer_v3": [ "0.0301973834223185", "0.0652880970289538", "0.303265329856317" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0226", "subject": "Probability", "topic": "Discrete distributions", "subtopic": "Poisson", "level": "3", "keywords": [ "probability", "Poisson" ], "problem_v1": "Cars arriving for gasoline at a Shell station follow a Poisson distribution with a mean of 9 per hour. A. Determine the probability that over the next hour, only one car will arrive. Probability=[ANS]\nB. Compute the probability that in the next 6 hours, more than 24 cars will arrive. Probability=[ANS]", "answer_v1": [ "0.00111068823678012", "0.999996217307645" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Cars arriving for gasoline at a Shell station follow a Poisson distribution with a mean of 4 per hour. A. Determine the probability that over the next hour, only one car will arrive. Probability=[ANS]\nB. Compute the probability that in the next 8 hours, more than 17 cars will arrive. Probability=[ANS]", "answer_v2": [ "0.0732625555549367", "0.997230940271691" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Cars arriving for gasoline at a Shell station follow a Poisson distribution with a mean of 6 per hour. A. Determine the probability that over the next hour, only one car will arrive. Probability=[ANS]\nB. Compute the probability that in the next 6 hours, more than 19 cars will arrive. Probability=[ANS]", "answer_v3": [ "0.0148725130599982", "0.998577930736881" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0228", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "3", "keywords": [ "Joint Distribution", "Uniform" ], "problem_v1": "Two points along a straight stick of length $45$ cm are randomly selected. The stick is then broken at those two points. Find the probability that all of the resulting pieces have lenght at least $7$ cm. probability=[ANS]", "answer_v1": [ "0.284444444444444" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two points along a straight stick of length $31$ cm are randomly selected. The stick is then broken at those two points. Find the probability that all of the resulting pieces have lenght at least $7.5$ cm. probability=[ANS]", "answer_v2": [ "0.0751821019771072" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two points along a straight stick of length $36$ cm are randomly selected. The stick is then broken at those two points. Find the probability that all of the resulting pieces have lenght at least $6$ cm. probability=[ANS]", "answer_v3": [ "0.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0229", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "3", "keywords": [ "Joint Distribution", "Uniform" ], "problem_v1": "A man and a woman agree to meet at a cafe about noon. If the man arrives at a time uniformly distributed between $11:40$ and $12:15$ and if the woman independently arrives at a time uniformly distributed between $11:55$ and $12:35$, what is the probability that the first to arrive waits no longer than $10$ minutes? [ANS]", "answer_v1": [ "0.285714285714286" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A man and a woman agree to meet at a cafe about noon. If the man arrives at a time uniformly distributed between $11:30$ and $12:15$ and if the woman independently arrives at a time uniformly distributed between $11:40$ and $12:55$, what is the probability that the first to arrive waits no longer than $5$ minutes? [ANS]", "answer_v2": [ "0.103703703703704" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A man and a woman agree to meet at a cafe about noon. If the man arrives at a time uniformly distributed between $11:30$ and $12:15$ and if the woman independently arrives at a time uniformly distributed between $11:45$ and $12:25$, what is the probability that the first to arrive waits no longer than $5$ minutes? [ANS]", "answer_v3": [ "0.166666666666667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0230", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "3", "keywords": [ "Joint Distribution", "Uniform" ], "problem_v1": "Two points are selected randomly on a line of length $34$ so as to be on opposite sides of the midpoint of the line. In other words, the two points $X$ and $Y$ are independent random variables such that $X$ is uniformly distributed over $[0,17)$ and $Y$ is uniformly distributed over $(17,34]$. Find the probability that the distance between the two points is greater than $10$.\nanswer: [ANS]", "answer_v1": [ "0.826989619377163" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two points are selected randomly on a line of length $12$ so as to be on opposite sides of the midpoint of the line. In other words, the two points $X$ and $Y$ are independent random variables such that $X$ is uniformly distributed over $[0,6)$ and $Y$ is uniformly distributed over $(6,12]$. Find the probability that the distance between the two points is greater than $5$.\nanswer: [ANS]", "answer_v2": [ "0.652777777777778" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two points are selected randomly on a line of length $20$ so as to be on opposite sides of the midpoint of the line. In other words, the two points $X$ and $Y$ are independent random variables such that $X$ is uniformly distributed over $[0,10)$ and $Y$ is uniformly distributed over $(10,20]$. Find the probability that the distance between the two points is greater than $6$.\nanswer: [ANS]", "answer_v3": [ "0.82" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0231", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "4", "keywords": [ "Joint Distribution", "Uniform", "Roots" ], "problem_v1": "Let $A$, $B$, and $C$ be independent random variables, uniformly distributed over $[0,8],$ $[0,16],$ and $[0,6]$ respectively. What is the probability that both roots of the equation $Ax^2+Bx+C=0$ are real? [ANS]", "answer_v1": [ "1-8/9 * sqrt(8*6)/16" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $A$, $B$, and $C$ be independent random variables, uniformly distributed over $[0,2],$ $[0,12],$ and $[0,3]$ respectively. What is the probability that both roots of the equation $Ax^2+Bx+C=0$ are real? [ANS]", "answer_v2": [ "1-8/9 * sqrt(2*3)/12" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $A$, $B$, and $C$ be independent random variables, uniformly distributed over $[0,4],$ $[0,12],$ and $[0,3]$ respectively. What is the probability that both roots of the equation $Ax^2+Bx+C=0$ are real? [ANS]", "answer_v3": [ "1-8/9 * sqrt(4*3)/12" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Probability_0232", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "3", "keywords": [ "Joint Distribution", "Uniform" ], "problem_v1": "$x$ and $y$ are uniformly distributed over the interval $[0,1].$ Find the probability that $|x-y|$, the distance between $x$ and $y$, is less than $0.7.$ [ANS]", "answer_v1": [ "0.91" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$x$ and $y$ are uniformly distributed over the interval $[0,1].$ Find the probability that $|x-y|$, the distance between $x$ and $y$, is less than $0.1.$ [ANS]", "answer_v2": [ "0.19" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$x$ and $y$ are uniformly distributed over the interval $[0,1].$ Find the probability that $|x-y|$, the distance between $x$ and $y$, is less than $0.3.$ [ANS]", "answer_v3": [ "0.51" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0233", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "2", "keywords": [ "probability", "uniform distribution", "statistics" ], "problem_v1": "The weather in Rochester in December is fairly constant. Records indicate that the low temperature for each day of the month tend to have a uniform distribution over the interval $15^{\\circ}$ to $35^{\\circ}$ F. A business man arrives on a randomly selected day in December.\n(a) What is the probability that the temperature will be above $27^{\\circ}$? answer: [ANS]\n(b) What is the probability that the temperature will be between $19^{\\circ}$ and $30^{\\circ}$? answer: [ANS]\n(c) What is the expected temperature? answer: [ANS]", "answer_v1": [ "0.4", "0.55", "25" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The weather in Rochester in December is fairly constant. Records indicate that the low temperature for each day of the month tend to have a uniform distribution over the interval $15^{\\circ}$ to $35^{\\circ}$ F. A business man arrives on a randomly selected day in December.\n(a) What is the probability that the temperature will be above $18^{\\circ}$? answer: [ANS]\n(b) What is the probability that the temperature will be between $20^{\\circ}$ and $26^{\\circ}$? answer: [ANS]\n(c) What is the expected temperature? answer: [ANS]", "answer_v2": [ "0.85", "0.3", "25" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The weather in Rochester in December is fairly constant. Records indicate that the low temperature for each day of the month tend to have a uniform distribution over the interval $15^{\\circ}$ to $35^{\\circ}$ F. A business man arrives on a randomly selected day in December.\n(a) What is the probability that the temperature will be above $21^{\\circ}$? answer: [ANS]\n(b) What is the probability that the temperature will be between $19^{\\circ}$ and $27^{\\circ}$? answer: [ANS]\n(c) What is the expected temperature? answer: [ANS]", "answer_v3": [ "0.7", "0.4", "25" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0234", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "2", "keywords": [ "probability", "uniform distribution", "statistics" ], "problem_v1": "Suppose the time to process a loan application follows a uniform distribution over the range $8$ to $15$ days. What is the probability that a randomly selected loan application takes longer than $12$ days to process? answer: [ANS]", "answer_v1": [ "0.428571428571429" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose the time to process a loan application follows a uniform distribution over the range $5$ to $17$ days. What is the probability that a randomly selected loan application takes longer than $7$ days to process? answer: [ANS]", "answer_v2": [ "0.833333333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose the time to process a loan application follows a uniform distribution over the range $6$ to $16$ days. What is the probability that a randomly selected loan application takes longer than $9$ days to process? answer: [ANS]", "answer_v3": [ "0.7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0235", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "2", "keywords": [ "probability", "uniform distribution", "statistics" ], "problem_v1": "Suppose $x$ is a random variable best described by a uniform probability that ranges from $2$ to $5$. Compute the following:\n(a) $\\ $ the probability density function $f(x)=$ [ANS]\n(b) $\\ $ the mean $\\mu=$ [ANS]\n(c) $\\ $ the standard deviation $\\sigma=$ [ANS]\n(d) $\\ $ $P(\\mu-\\sigma \\leq x \\leq \\mu+\\sigma)=$ [ANS]\n(e) $\\ $ $P(x \\geq 3.86)=$ [ANS]", "answer_v1": [ "0.333333333333333", "3.5", "0.866025403784439", "0.577350269189626", "0.38" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Suppose $x$ is a random variable best described by a uniform probability that ranges from $0$ to $6$. Compute the following:\n(a) $\\ $ the probability density function $f(x)=$ [ANS]\n(b) $\\ $ the mean $\\mu=$ [ANS]\n(c) $\\ $ the standard deviation $\\sigma=$ [ANS]\n(d) $\\ $ $P(\\mu-\\sigma \\leq x \\leq \\mu+\\sigma)=$ [ANS]\n(e) $\\ $ $P(x \\geq 0.9)=$ [ANS]", "answer_v2": [ "0.166666666666667", "3", "1.73205080756888", "0.577350269189626", "0.85" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Suppose $x$ is a random variable best described by a uniform probability that ranges from $0$ to $5$. Compute the following:\n(a) $\\ $ the probability density function $f(x)=$ [ANS]\n(b) $\\ $ the mean $\\mu=$ [ANS]\n(c) $\\ $ the standard deviation $\\sigma=$ [ANS]\n(d) $\\ $ $P(\\mu-\\sigma \\leq x \\leq \\mu+\\sigma)=$ [ANS]\n(e) $\\ $ $P(x \\geq 1.4)=$ [ANS]", "answer_v3": [ "0.2", "2.5", "1.44337567297406", "0.577350269189626", "0.72" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Probability_0236", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "4", "keywords": [ "Uniform Distribution", "Roots" ], "problem_v1": "If $a$ is uniformly distributed over $[-23, 28]$, what is the probability that the roots of the equation x^2+ax+a+63=0 are both real? [ANS]", "answer_v1": [ "0.372549019607843" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $a$ is uniformly distributed over $[-16, 9]$, what is the probability that the roots of the equation x^2+ax+a+3=0 are both real? [ANS]", "answer_v2": [ "0.68" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $a$ is uniformly distributed over $[-16, 15]$, what is the probability that the roots of the equation x^2+ax+a+15=0 are both real? [ANS]", "answer_v3": [ "0.483870967741935" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0237", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "5", "keywords": [ "area" ], "problem_v1": "The VIP cafeteria door on the Death Star promptly opens at 11:00 am and closes at 1:00 pm (Standard Galactic Time). Nobody is allowed to enter at other times but guests can stay until they finish their meal. To keep their lean physiques, Sith Lords usually spend their allotted 14-minute lunch break in the cafeteria sipping organic kale smoothies. Darth Sidious has a yoga class at 11:00 am, so he never has lunch before noon. Darth Vader must use a straw, so he is allowed an additional 6 minutes to slurp his smoothie. What is the probability that the two of them meet today in the cafeteria? $P(\\text{meet})$=[ANS]", "answer_v1": [ "[7200-(60-14)*(60-14)/2-(60-20+120-20)*60/2]/7200" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The VIP cafeteria door on the Death Star promptly opens at 11:00 am and closes at 1:00 pm (Standard Galactic Time). Nobody is allowed to enter at other times but guests can stay until they finish their meal. To keep their lean physiques, Sith Lords usually spend their allotted 10-minute lunch break in the cafeteria sipping organic kale smoothies. Darth Sidious has a yoga class at 11:00 am, so he never has lunch before noon. Darth Vader must use a straw, so he is allowed an additional 8 minutes to slurp his smoothie. What is the probability that the two of them meet today in the cafeteria? $P(\\text{meet})$=[ANS]", "answer_v2": [ "[7200-(60-10)*(60-10)/2-(60-18+120-18)*60/2]/7200" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The VIP cafeteria door on the Death Star promptly opens at 11:00 am and closes at 1:00 pm (Standard Galactic Time). Nobody is allowed to enter at other times but guests can stay until they finish their meal. To keep their lean physiques, Sith Lords usually spend their allotted 11-minute lunch break in the cafeteria sipping organic kale smoothies. Darth Sidious has a yoga class at 11:00 am, so he never has lunch before noon. Darth Vader must use a straw, so he is allowed an additional 6 minutes to slurp his smoothie. What is the probability that the two of them meet today in the cafeteria? $P(\\text{meet})$=[ANS]", "answer_v3": [ "[7200-(60-11)*(60-11)/2-(60-17+120-17)*60/2]/7200" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0238", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "2", "keywords": [ "probability", "continuous", "uniform", "distribution" ], "problem_v1": "Suppose that random variable $X$ is uniformly distributed between 7 and 27. Draw a graph of the density function, and then use it to help find the following probabilities: A. $P(X > 27)$=[ANS]\nB. $P(X < 15.7)$=[ANS]\nC. $P(11 < X < 24)$=[ANS]\nD. $P(14 < X < 31)$=[ANS]", "answer_v1": [ "0", "0.435", "0.65", "0.65" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose that random variable $X$ is uniformly distributed between 4 and 30. Draw a graph of the density function, and then use it to help find the following probabilities: A. $P(X > 30)$=[ANS]\nB. $P(X < 8.3)$=[ANS]\nC. $P(7 < X < 25)$=[ANS]\nD. $P(11 < X < 32)$=[ANS]", "answer_v2": [ "0", "0.165384615384615", "0.692307692307692", "0.730769230769231" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose that random variable $X$ is uniformly distributed between 5 and 27. Draw a graph of the density function, and then use it to help find the following probabilities: A. $P(X > 27)$=[ANS]\nB. $P(X < 10.5)$=[ANS]\nC. $P(9 < X < 25)$=[ANS]\nD. $P(12 < X < 32)$=[ANS]", "answer_v3": [ "0", "0.25", "0.727272727272727", "0.681818181818182" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0239", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Uniform", "level": "2", "keywords": [ "probability", "continuous", "uniform", "distribution" ], "problem_v1": "The amount of time it takes for a student to complete a statistics quiz is uniformly distributed (or, given by a random variable that is uniformly distributed) between 33 and 61 minutes. One student is selected at random. Find the probability of the following events. A. The student requires more than 56 minutes to complete the quiz. Probability=[ANS]\nB. The student completes the quiz in a time between 39 and 42 minutes. Probability=[ANS]\nC. The student completes the quiz in exactly 41.65 minutes. Probability=[ANS]", "answer_v1": [ "(61-56)*1/(61-33)", "(42-39)*1/(61-33)", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The amount of time it takes for a student to complete a statistics quiz is uniformly distributed (or, given by a random variable that is uniformly distributed) between 25 and 65 minutes. One student is selected at random. Find the probability of the following events. A. The student requires more than 61 minutes to complete the quiz. Probability=[ANS]\nB. The student completes the quiz in a time between 30 and 36 minutes. Probability=[ANS]\nC. The student completes the quiz in exactly 41.58 minutes. Probability=[ANS]", "answer_v2": [ "(65-61)*1/(65-25)", "(36-30)*1/(65-25)", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The amount of time it takes for a student to complete a statistics quiz is uniformly distributed (or, given by a random variable that is uniformly distributed) between 28 and 61 minutes. One student is selected at random. Find the probability of the following events. A. The student requires more than 57 minutes to complete the quiz. Probability=[ANS]\nB. The student completes the quiz in a time between 33 and 37 minutes. Probability=[ANS]\nC. The student completes the quiz in exactly 41.73 minutes. Probability=[ANS]", "answer_v3": [ "(61-57)*1/(61-28)", "(37-33)*1/(61-28)", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0240", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "3", "keywords": [ "Exponential Distribution", "Hazard Rate" ], "problem_v1": "Suppose that the life distribution of an item has hazard rate function $\\lambda(t)=3.9 t^2, \\ t>0$. What is the probability that\n(a) the item doesn't survive to age $2$? [ANS]\n(b) the item's lifetime is between $0.5$ and $2$? [ANS]\n(c) a $1$-year-old item will survive to age $3.5$? [ANS]", "answer_v1": [ "0.999969567516992", "0.84998565774239", "2.28083643205853E-24" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that the life distribution of an item has hazard rate function $\\lambda(t)=1.5 t^2, \\ t>0$. What is the probability that\n(a) the item survives to age $3$? [ANS]\n(b) the item's lifetime is between $1.5$ and $2$? [ANS]\n(c) a $0.5$-year-old item will survive to age $2.5$? [ANS]", "answer_v2": [ "1.37095908638409E-06", "0.16666576101857", "0.000430742540575688" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that the life distribution of an item has hazard rate function $\\lambda(t)=2.4 t^2, \\ t>0$. What is the probability that\n(a) the item doesn't survive to age $2$? [ANS]\n(b) the item's lifetime is between $0.5$ and $3$? [ANS]\n(c) a $0.5$-year-old item will survive to age $3$? [ANS]", "answer_v3": [ "0.998338442726826", "0.90483741761982", "4.59905537865232E-10" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0241", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "3", "keywords": [ "Exponential Distribution", "Random Variable", "Probability Density Function", "PDF" ], "problem_v1": "Let $X$ be an exponential random variable with parameter $\\lambda=8$, and let $Y$ be the random variable defined by $Y=7 e^X$. Compute the probability density function of $Y$: $f_Y(t)=$ [ANS]", "answer_v1": [ "8 * (7^8) * t^{-9}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $X$ be an exponential random variable with parameter $\\lambda=2$, and let $Y$ be the random variable defined by $Y=10 e^X$. Compute the probability density function of $Y$: $f_Y(t)=$ [ANS]", "answer_v2": [ "2 * (10^2) * t^{-3}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $X$ be an exponential random variable with parameter $\\lambda=4$, and let $Y$ be the random variable defined by $Y=7 e^X$. Compute the probability density function of $Y$: $f_Y(t)=$ [ANS]", "answer_v3": [ "4 * (7^4) * t^{-5}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Probability_0242", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "3", "keywords": [ "Exponential Distribution", "Random Variable" ], "problem_v1": "Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter $\\lambda=0.7$. What is\n(a) the probability that a repair takes less than $7$ hours? [ANS]\n(b) the conditional probability that a repair takes at least $10$ hours, given that it takes more than $8$ hours? [ANS]", "answer_v1": [ "0.992553416929076", "0.246596963941606" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter $\\lambda=0.1$. What is\n(a) the probability that a repair time exceeds $10$ hours? [ANS]\n(b) the conditional probability that a repair takes at least $10$ hours, given that it takes more than $5$ hours? [ANS]", "answer_v2": [ "0.367879441171442", "0.606530659712633" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter $\\lambda=0.3$. What is\n(a) the probability that a repair time exceeds $7$ hours? [ANS]\n(b) the conditional probability that a repair takes at least $8$ hours, given that it takes more than $6$ hours? [ANS]", "answer_v3": [ "0.122456428252982", "0.548811636094026" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0243", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "3", "keywords": [ "Random variables", "Exponential distribution", "determination of probabilities for an Exponential variable over given ranges", "finding a conditional probability" ], "problem_v1": "The time (in minutes) between arrivals of customers to a post office is to be modelled by the Exponential distribution with mean $0.65$. Please give your answers to two decimal places.\nPart a) What is the probability that the time between consecutive customers is less than 15 seconds? [ANS]\nPart b) Find the probability that the time between consecutive customers is between ten and fifteen seconds. [ANS]\nPart c) Given that the time between consecutive customers arriving is greater than ten seconds, what is the chance that it is greater than fifteen seconds? [ANS]", "answer_v1": [ "0.32", "0.09", "0.88" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The time (in minutes) between arrivals of customers to a post office is to be modelled by the Exponential distribution with mean $0.36$. Please give your answers to two decimal places.\nPart a) What is the probability that the time between consecutive customers is less than 15 seconds? [ANS]\nPart b) Find the probability that the time between consecutive customers is between ten and fifteen seconds. [ANS]\nPart c) Given that the time between consecutive customers arriving is greater than ten seconds, what is the chance that it is greater than fifteen seconds? [ANS]", "answer_v2": [ "0.5", "0.13", "0.79" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The time (in minutes) between arrivals of customers to a post office is to be modelled by the Exponential distribution with mean $0.46$. Please give your answers to two decimal places.\nPart a) What is the probability that the time between consecutive customers is less than 15 seconds? [ANS]\nPart b) Find the probability that the time between consecutive customers is between ten and fifteen seconds. [ANS]\nPart c) Given that the time between consecutive customers arriving is greater than ten seconds, what is the chance that it is greater than fifteen seconds? [ANS]", "answer_v3": [ "0.42", "0.12", "0.83" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0244", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "3", "keywords": [ "probability", "continuous", "exponential", "distribution" ], "problem_v1": "The manager of a supermarket tracked the amount of time needed for customers to be served by the cashier. After checking with his statistics professor, he concluded that the checkout times are exponentially distributed with a mean of 6.5 minutes. What propotion of customers require more than 11 minutes to check out? Proportion=[ANS]", "answer_v1": [ "0.184094200653304" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The manager of a supermarket tracked the amount of time needed for customers to be served by the cashier. After checking with his statistics professor, he concluded that the checkout times are exponentially distributed with a mean of 5 minutes. What propotion of customers require more than 12 minutes to check out? Proportion=[ANS]", "answer_v2": [ "0.0907179532894125" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The manager of a supermarket tracked the amount of time needed for customers to be served by the cashier. After checking with his statistics professor, he concluded that the checkout times are exponentially distributed with a mean of 5.5 minutes. What propotion of customers require more than 11 minutes to check out? Proportion=[ANS]", "answer_v3": [ "0.135335283236613" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0245", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "3", "keywords": [ "probability", "continuous", "exponential", "distribution" ], "problem_v1": "The manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 7.5 minutes. What is the probability that a car can complete the transaction in less than 5 minutes? Probability=[ANS]", "answer_v1": [ "0.486582880967408" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 6 minutes. What is the probability that a car can complete the transaction in less than 6 minutes? Probability=[ANS]", "answer_v2": [ "0.632120558828558" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 6.5 minutes. What is the probability that a car can complete the transaction in less than 5 minutes? Probability=[ANS]", "answer_v3": [ "0.536630630768825" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0246", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "3", "keywords": [ "probability", "continuous", "exponential", "distribution" ], "problem_v1": "A bank wishing to increase its customer base advertises that it has the fastest service and that virtually all of its customers are served in less than 10 minutes. A management scientist has studied the service times and concluded that service times are exponentially distributed with a mean of 5.5 minutes. Determine what the bank means when it claims 'virtually all' its customers are served in under 10 minutes. Proportion of customers served in under 10 minutes=[ANS]", "answer_v1": [ "0.837679388818152" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A bank wishing to increase its customer base advertises that it has the fastest service and that virtually all of its customers are served in less than 11 minutes. A management scientist has studied the service times and concluded that service times are exponentially distributed with a mean of 4 minutes. Determine what the bank means when it claims 'virtually all' its customers are served in under 11 minutes. Proportion of customers served in under 11 minutes=[ANS]", "answer_v2": [ "0.936072138793292" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A bank wishing to increase its customer base advertises that it has the fastest service and that virtually all of its customers are served in less than 10 minutes. A management scientist has studied the service times and concluded that service times are exponentially distributed with a mean of 4.5 minutes. Determine what the bank means when it claims 'virtually all' its customers are served in under 10 minutes. Proportion of customers served in under 10 minutes=[ANS]", "answer_v3": [ "0.891631976778104" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0247", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Exponential", "level": "2", "keywords": [ "probability", "continuous", "exponential", "distribution" ], "problem_v1": "Suppose that $X$ is an exponentially distributed random variable with $\\lambda=0.63$. Find each of the following probabilities: A. $P(X > 1)$=[ANS]\nB. $P(X > 0.49)$=[ANS]\nC. $P(X < 0.63)$=[ANS]\nD. $P(0.49 < X < 3.31)$=[ANS]", "answer_v1": [ "0.532591801006897", "0.73440105729871", "0.327598737602997", "0.610131223675719" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose that $X$ is an exponentially distributed random variable with $\\lambda=0.29$. Find each of the following probabilities: A. $P(X > 1)$=[ANS]\nB. $P(X > 0.1)$=[ANS]\nC. $P(X < 0.29)$=[ANS]\nD. $P(0.22 < X < 1.55)$=[ANS]", "answer_v2": [ "0.748263567578565", "0.971416464466605", "0.0806606824334818", "0.300245573877598" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose that $X$ is an exponentially distributed random variable with $\\lambda=0.41$. Find each of the following probabilities: A. $P(X > 1)$=[ANS]\nB. $P(X > 0.27)$=[ANS]\nC. $P(X < 0.41)$=[ANS]\nD. $P(0.32 < X < 2.18)$=[ANS]", "answer_v3": [ "0.663650250136319", "0.895207270829981", "0.154730696472182", "0.467944126304949" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0248", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "Normal Distribution", "z-score" ], "problem_v1": "Suppose the random variable $x$ is best described by a normal distribution with $\\mu=28$ and $\\sigma=6.7$. Find the $z$-score that corresponds to each of the following $x$ values.\n(a) $x=29$ $z=$ [ANS]\n(b) $x=32$ $z=$ [ANS]\n(c) $x=19$ $z=$ [ANS]\n(d) $x=20$ $z=$ [ANS]\n(e) $x=27$ $z=$ [ANS]\n(f) $x=27$ $z=$ [ANS]", "answer_v1": [ "0.149253731343284", "0.597014925373134", "-1.34328358208955", "-1.19402985074627", "-0.149253731343284", "-0.149253731343284" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Suppose the random variable $x$ is best described by a normal distribution with $\\mu=20$ and $\\sigma=9.5$. Find the $z$-score that corresponds to each of the following $x$ values.\n(a) $x=14$ $z=$ [ANS]\n(b) $x=20$ $z=$ [ANS]\n(c) $x=39$ $z=$ [ANS]\n(d) $x=19$ $z=$ [ANS]\n(e) $x=15$ $z=$ [ANS]\n(f) $x=20$ $z=$ [ANS]", "answer_v2": [ "-0.631578947368421", "0", "2", "-0.105263157894737", "-0.526315789473684", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Suppose the random variable $x$ is best described by a normal distribution with $\\mu=23$ and $\\sigma=6.9$. Find the $z$-score that corresponds to each of the following $x$ values.\n(a) $x=18$ $z=$ [ANS]\n(b) $x=27$ $z=$ [ANS]\n(c) $x=16$ $z=$ [ANS]\n(d) $x=20$ $z=$ [ANS]\n(e) $x=35$ $z=$ [ANS]\n(f) $x=38$ $z=$ [ANS]", "answer_v3": [ "-0.72463768115942", "0.579710144927536", "-1.01449275362319", "-0.434782608695652", "1.73913043478261", "2.17391304347826" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0249", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation" ], "problem_v1": "For a normal distribution, find the percentage of data that are\n(a) Within 2.51 standard deviations of the mean. [ANS] \\% (b) Between 3 standard deviations below the mean and 2 standard deviations above the mean [ANS] \\% (c) Less than $\\mu-2 \\sigma$ [ANS] \\%", "answer_v1": [ "98.7926883839206", "97.5899970020215", "2.27501309615917" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "For a normal distribution, find the percentage of data that are\n(a) Within 1 standard deviation of the mean. [ANS] \\% (b) Greater than 2 standard deviations below the mean [ANS] \\% (c) Less than $\\mu-0.5 \\sigma$ [ANS] \\%", "answer_v2": [ "68.2689492137085", "97.7249867038964", "30.8537537739272" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "For a normal distribution, find the percentage of data that are\n(a) Within 1 standard deviation of the mean. [ANS] \\% (b) Greater than 2.5 standard deviations below the mean [ANS] \\% (c) Less than $\\mu-3 \\sigma$ [ANS] \\%", "answer_v3": [ "68.2689492137085", "99.3790333665095", "0.134989704504245" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0250", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "Normal Distribution", "Random Variable" ], "problem_v1": "Find the value of the standard normal random variable $z$, called $z_0$ such that:\n(a) $P(z \\leq z_0)=0.8764$ $z_0=$ [ANS]\n(b) $P(-z_0 \\leq z \\leq z_0)=0.5816$ $z_0=$ [ANS]\n(c) $P(-z_0 \\leq z \\leq z_0)=0.6208$ $z_0=$ [ANS]\n(d) $P(z \\geq z_0)=0.1387$ $z_0=$ [ANS]\n(e) $P(-z_0 \\leq z \\leq 0)=0.1508$ $z_0=$ [ANS]\n(f) $P(-1.57 \\leq z \\leq z_0)=0.6073$ $z_0=$ [ANS]", "answer_v1": [ "1.15717712873752", "0.809200199846154", "0.879371278085369", "1.08617856202221", "0.387481200344059", "0.42752085284594" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the value of the standard normal random variable $z$, called $z_0$ such that:\n(a) $P(z \\leq z_0)=0.5415$ $z_0=$ [ANS]\n(b) $P(-z_0 \\leq z \\leq z_0)=0.9318$ $z_0=$ [ANS]\n(c) $P(-z_0 \\leq z \\leq z_0)=0.1494$ $z_0=$ [ANS]\n(d) $P(z \\geq z_0)=0.333$ $z_0=$ [ANS]\n(e) $P(-z_0 \\leq z \\leq 0)=0.4734$ $z_0=$ [ANS]\n(f) $P(-2.07 \\leq z \\leq z_0)=0.6391$ $z_0=$ [ANS]", "answer_v2": [ "0.104213400269085", "1.82368306868179", "0.188352924440269", "0.431644239383956", "1.93329392323534", "0.407827939358199" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the value of the standard normal random variable $z$, called $z_0$ such that:\n(a) $P(z \\leq z_0)=0.6567$ $z_0=$ [ANS]\n(b) $P(-z_0 \\leq z \\leq z_0)=0.6054$ $z_0=$ [ANS]\n(c) $P(-z_0 \\leq z \\leq z_0)=0.2786$ $z_0=$ [ANS]\n(d) $P(z \\geq z_0)=0.2257$ $z_0=$ [ANS]\n(e) $P(-z_0 \\leq z \\leq 0)=0.1031$ $z_0=$ [ANS]\n(f) $P(-1.25 \\leq z \\leq z_0)=0.5674$ $z_0=$ [ANS]", "answer_v3": [ "0.403473399026278", "0.851304906640506", "0.35658835031219", "0.753083065467299", "0.261379324048839", "0.448212281456609" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Probability_0251", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "Find the z-score from a standard normal distribution that satisfies each of the following statements.\n(a) The point z with 2.09 percent of the observations falling below it. $z=$ [ANS]\n(b) The closest point z with 26.85 percent of the observations falling above it. $z=$ [ANS]", "answer_v1": [ "-2.03551", "0.617356" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the z-score from a standard normal distribution that satisfies each of the following statements.\n(a) The point z with 1.25 percent of the observations falling below it. $z=$ [ANS]\n(b) The closest point z with 42.89 percent of the observations falling above it. $z=$ [ANS]", "answer_v2": [ "-2.2414", "0.179175" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the z-score from a standard normal distribution that satisfies each of the following statements.\n(a) The point z with 11.51 percent of the observations falling below it. $z=$ [ANS]\n(b) The closest point z with 27.94 percent of the observations falling above it. $z=$ [ANS]", "answer_v3": [ "-1.19984", "0.584625" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0252", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "3", "keywords": [ "Gaussian' 'distribution' 'integral" ], "problem_v1": "Find the normalizing constant $c$ so that $\\int_{-\\infty}^{\\infty}ce^{-x^2/8} \\, dx=1$. $c=$ [ANS]", "answer_v1": [ "0.199471140200716" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the normalizing constant $c$ so that $\\int_{-\\infty}^{\\infty}ce^{-x^2/2} \\, dx=1$. $c=$ [ANS]", "answer_v2": [ "0.398942280401433" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the normalizing constant $c$ so that $\\int_{-\\infty}^{\\infty}ce^{-x^2/4} \\, dx=1$. $c=$ [ANS]", "answer_v3": [ "0.282094791773878" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0253", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "The U.S. Bureau of the Census conducts nationwide sureys on characteristics of U.S. households.\n$\\begin{array}{cc}\\hline Household size & Relative Frequency \\\\ \\hline 1 & 0.075 \\\\ \\hline 2 & 0.2 \\\\ \\hline 3 & 0.225 \\\\ \\hline 4 & 0.325 \\\\ \\hline 5 & 0.125 \\\\ \\hline 6 & 0.025 \\\\ \\hline 7 & 0.025 \\\\ \\hline Total & 1 \\\\ \\hline \\end{array}$\na) Use the previous relative frequency distribution to obtain the percentage of U.S. households that are between sizes 3 and 5. answer: [ANS]\nb) Use your answer from part a) to estimate the area under the corresponding normal curve that lies between 3 and 5. answer: [ANS]", "answer_v1": [ "67.5", "0.675" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The U.S. Bureau of the Census conducts nationwide sureys on characteristics of U.S. households.\n$\\begin{array}{cc}\\hline Household size & Relative Frequency \\\\ \\hline 1 & 0.2 \\\\ \\hline 2 & 0.125 \\\\ \\hline 3 & 0.275 \\\\ \\hline 4 & 0.175 \\\\ \\hline 5 & 0.075 \\\\ \\hline 6 & 0.05 \\\\ \\hline 7 & 0.1 \\\\ \\hline Total & 1 \\\\ \\hline \\end{array}$\na) Use the previous relative frequency distribution to obtain the percentage of U.S. households that are between sizes 3 and 5. answer: [ANS]\nb) Use your answer from part a) to estimate the area under the corresponding normal curve that lies between 3 and 5. answer: [ANS]", "answer_v2": [ "52.5", "0.525" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The U.S. Bureau of the Census conducts nationwide sureys on characteristics of U.S. households.\n$\\begin{array}{cc}\\hline Household size & Relative Frequency \\\\ \\hline 1 & 0.1 \\\\ \\hline 2 & 0.15 \\\\ \\hline 3 & 0.125 \\\\ \\hline 4 & 0.275 \\\\ \\hline 5 & 0.2 \\\\ \\hline 6 & 0.075 \\\\ \\hline 7 & 0.075 \\\\ \\hline Total & 1 \\\\ \\hline \\end{array}$\na) Use the previous relative frequency distribution to obtain the percentage of U.S. households that are between sizes 3 and 5. answer: [ANS]\nb) Use your answer from part a) to estimate the area under the corresponding normal curve that lies between 3 and 5. answer: [ANS]", "answer_v3": [ "60", "0.6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0254", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "Statistics", "Probability", "Normal Distribution" ], "problem_v1": "Determine the following z-scores:\n(a) $z_{0.15}$=[ANS]\n(b) $z_{0.25}$=[ANS]", "answer_v1": [ "1.03643338949379", "0.674489750196071" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine the following z-scores:\n(a) $z_{0.5}$=[ANS]\n(b) $z_{0.05}$=[ANS]", "answer_v2": [ "0", "1.64485362695934" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine the following z-scores:\n(a) $z_{0.35}$=[ANS]\n(b) $z_{0.2}$=[ANS]", "answer_v3": [ "0.385320466407568", "0.841621233572914" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0255", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "As reported in \"Runner's World\" magazine, the times of the finishers in the New York City 10 km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. Let x denote finishing time for the finishers.\na) The distribution of the variable x has mean [ANS] and standard deviation [ANS].\nb) The distribution of the standardized variable z has mean [ANS] and standard deviation [ANS].\nc) The percentage of finishers with times between 55 and 80 minutes is equal to the area under the standard normal curve between [ANS] and [ANS].\nd) The percentage of finishers with times exceeding 81 minutes is equal to the area under the standard normal curve that lies to the [ANS] of [ANS].", "answer_v1": [ "61", "9", "0", "1", "-0.666666666666667", "2.11111111111111", "right", "2.22222222222222" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "OE", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "As reported in \"Runner's World\" magazine, the times of the finishers in the New York City 10 km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. Let x denote finishing time for the finishers.\na) The distribution of the variable x has mean [ANS] and standard deviation [ANS].\nb) The distribution of the standardized variable z has mean [ANS] and standard deviation [ANS].\nc) The percentage of finishers with times between 35 and 90 minutes is equal to the area under the standard normal curve between [ANS] and [ANS].\nd) The percentage of finishers with times exceeding 68 minutes is equal to the area under the standard normal curve that lies to the [ANS] of [ANS].", "answer_v2": [ "61", "9", "0", "1", "-2.88888888888889", "3.22222222222222", "right", "0.777777777777778" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "OE", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "As reported in \"Runner's World\" magazine, the times of the finishers in the New York City 10 km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. Let x denote finishing time for the finishers.\na) The distribution of the variable x has mean [ANS] and standard deviation [ANS].\nb) The distribution of the standardized variable z has mean [ANS] and standard deviation [ANS].\nc) The percentage of finishers with times between 40 and 80 minutes is equal to the area under the standard normal curve between [ANS] and [ANS].\nd) The percentage of finishers with times exceeding 72 minutes is equal to the area under the standard normal curve that lies to the [ANS] of [ANS].", "answer_v3": [ "61", "9", "0", "1", "-2.33333333333333", "2.11111111111111", "right", "1.22222222222222" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "OE", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Probability_0256", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "Statistics" ], "problem_v1": "a) Fact: The region under the standard normal curve that lies to the left of 1.58 has area 0.9429. Without consulting a table or a calculator giving areas under the standard normal curve, determine the area under the standard normal curve that lies to the right of 1.58. answer: [ANS]\nb) Which property of the standard normal curve allowed you to answer part a)? [ANS] A. The total area under the curve is 1 B. Almost all the area under the standard normal curve lies between $-3$ and $3$ C. The standard normal curve extends indefinitely in both directions D. The standard normal curve is symmetric about 0 E. None of the above", "answer_v1": [ "0.0571", "A" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "a) Fact: The region under the standard normal curve that lies to the left of 1.94 has area 0.9738. Without consulting a table or a calculator giving areas under the standard normal curve, determine the area under the standard normal curve that lies to the right of 1.94. answer: [ANS]\nb) Which property of the standard normal curve allowed you to answer part a)? [ANS] A. The total area under the curve is 1 B. Almost all the area under the standard normal curve lies between $-3$ and $3$ C. The standard normal curve extends indefinitely in both directions D. The standard normal curve is symmetric about 0 E. None of the above", "answer_v2": [ "0.0262", "A" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "a) Fact: The region under the standard normal curve that lies to the left of 1.61 has area 0.9463. Without consulting a table or a calculator giving areas under the standard normal curve, determine the area under the standard normal curve that lies to the right of 1.61. answer: [ANS]\nb) Which property of the standard normal curve allowed you to answer part a)? [ANS] A. The total area under the curve is 1 B. The standard normal curve extends indefinitely in both directions C. Almost all the area under the standard normal curve lies between $-3$ and $3$ D. The standard normal curve is symmetric about 0 E. None of the above", "answer_v3": [ "0.0537", "A" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0257", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [], "problem_v1": "Consider a normal distribution curve where 80-th percentile is at 18 and the 35-th percentile is at 6. Use this information to find the mean, $\\mu$, and the standard deviation, $\\sigma$, of the distribution. a) $\\mu=$ [ANS]\nb) $\\sigma=$ [ANS]", "answer_v1": [ "9.76859438143179", "9.78041580964352" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider a normal distribution curve where 60-th percentile is at 14 and the 5-th percentile is at 10. Use this information to find the mean, $\\mu$, and the standard deviation, $\\sigma$, of the distribution. a) $\\mu=$ [ANS]\nb) $\\sigma=$ [ANS]", "answer_v2": [ "13.4661321131764", "2.10725869850431" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider a normal distribution curve where 65-th percentile is at 16 and the 15-th percentile is at 7. Use this information to find the mean, $\\mu$, and the standard deviation, $\\sigma$, of the distribution. a) $\\mu=$ [ANS]\nb) $\\sigma=$ [ANS]", "answer_v3": [ "13.5608406593914", "6.33020966508594" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0258", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "z-score", "normal distribution" ], "problem_v1": "Final exam scores in a mathematics course are normally distributed with a mean of 75 and a standard deviation of 11. Based on the above information and a Z-table, fill in the blanks in the table below.\nPrecision and other notes: (1) Percentiles should be recorded in percentage form to three decimal places. (2) Note that this problem does not use the rough values of the 68-95-99.7 rule (that is, the empirical rule); instead you must use more precise Z-table values for percentiles.\n$\\begin{array}{ccc}\\hline Exam score & Z-score & Percentile \\\\ \\hline 53 & [ANS] & [ANS] \\\\ \\hline 42 & [ANS] & [ANS] \\\\ \\hline [ANS] & 0.67 & [ANS] \\\\ \\hline [ANS] & [ANS] & 15.87 \\\\ \\hline \\end{array}$", "answer_v1": [ "-2", "2.27501309615917", "-3", "0.134989704504245", "82.37", "74.8571103920678", "64", "-1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Final exam scores in a mathematics course are normally distributed with a mean of 61 and a standard deviation of 15. Based on the above information and a Z-table, fill in the blanks in the table below.\nPrecision and other notes: (1) Percentiles should be recorded in percentage form to three decimal places. (2) Note that this problem does not use the rough values of the 68-95-99.7 rule (that is, the empirical rule); instead you must use more precise Z-table values for percentiles.\n$\\begin{array}{ccc}\\hline Exam score & Z-score & Percentile \\\\ \\hline 71.05 & [ANS] & [ANS] \\\\ \\hline 76 & [ANS] & [ANS] \\\\ \\hline [ANS] & 3 & [ANS] \\\\ \\hline [ANS] & [ANS] & 15.87 \\\\ \\hline \\end{array}$", "answer_v2": [ "0.67", "74.8571103920678", "1", "84.1344745086064", "106", "99.8650101019265", "46", "-1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Final exam scores in a mathematics course are normally distributed with a mean of 66 and a standard deviation of 11. Based on the above information and a Z-table, fill in the blanks in the table below.\nPrecision and other notes: (1) Percentiles should be recorded in percentage form to three decimal places. (2) Note that this problem does not use the rough values of the 68-95-99.7 rule (that is, the empirical rule); instead you must use more precise Z-table values for percentiles.\n$\\begin{array}{ccc}\\hline Exam score & Z-score & Percentile \\\\ \\hline 55 & [ANS] & [ANS] \\\\ \\hline 44 & [ANS] & [ANS] \\\\ \\hline [ANS] & 0.67 & [ANS] \\\\ \\hline [ANS] & [ANS] & 84.13 \\\\ \\hline \\end{array}$", "answer_v3": [ "-1", "15.8655252944872", "-2", "2.27501309615917", "73.37", "74.8571103920678", "77", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Probability_0259", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "statistics", "introduction", "normal distribution" ], "problem_v1": "Which of the following are true about all normal distributions? Check all that apply [ANS] A. They are symmetric. B. They have no major outliers. C. They have one peak. D. They are defined by the mean and standard deviation.\nThe z-score corresponding to an observed value of a variable tells you the number of standard deviations that the observation is from the mean [ANS] A. True B. False\nA positive z-score indicates that the observation is [ANS] A. above the mean B. below the mean", "answer_v1": [ "ABCD", "A", "A" ], "answer_type_v1": [ "MCM", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v2": "Which of the following are true about all normal distributions? Check all that apply [ANS] A. They have one peak. B. They have one large tail. C. They are all centered at zero. D. They are bimodal.\nThe z-score corresponding to an observed value of a variable tells you the number of standard deviations that the observation is from the mean [ANS] A. True B. False\nA positive z-score indicates that the observation is [ANS] A. above the mean B. below the mean", "answer_v2": [ "A", "A", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v3": "Which of the following are true about all normal distributions? Check all that apply [ANS] A. They have no major outliers. B. They have one large tail. C. They are bimodal. D. They have one peak.\nThe z-score corresponding to an observed value of a variable tells you the number of standard deviations that the observation is from the mean [ANS] A. True B. False\nA positive z-score indicates that the observation is [ANS] A. above the mean B. below the mean", "answer_v3": [ "AD", "A", "A" ], "answer_type_v3": [ "MCM", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B" ], [ "A", "B" ] ] }, { "id": "Probability_0260", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "5", "keywords": [], "problem_v1": "Let $X$ be normally distributed with mean, $\\mu$, and standard deviation, $\\sigma$=$\\mu$. Also suppose $\\mbox{Pr}(-7 < X < 17)=0.3499$. \u00a0 Find the value of the mean, $\\mu$. $\\mu=$ [ANS]", "answer_v1": [ "20.1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $X$ be normally distributed with mean, $\\mu$, and standard deviation, $\\sigma$=$\\mu$. Also suppose $\\mbox{Pr}(-10 < X < 12)=0.7874$. \u00a0 Find the value of the mean, $\\mu$. $\\mu=$ [ANS]", "answer_v2": [ "6.6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $X$ be normally distributed with mean, $\\mu$, and standard deviation, $\\sigma$=$\\mu$. Also suppose $\\mbox{Pr}(-7 < X < 13)=0.5071$. \u00a0 Find the value of the mean, $\\mu$. $\\mu=$ [ANS]", "answer_v3": [ "11.3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0261", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "3", "keywords": [ "empirical rule" ], "problem_v1": "Suppose the scores of students on an exam are Normally distributed with a mean of 489 and a standard deviation of 71. Then approximately 99.7\\% of the exam scores lie between the numbers [ANS] and [ANS] such that the mean is halfway between these two integers.", "answer_v1": [ "276", "702" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose the scores of students on an exam are Normally distributed with a mean of 217 and a standard deviation of 53. Then approximately 99.7\\% of the exam scores lie between the numbers [ANS] and [ANS] such that the mean is halfway between these two integers.", "answer_v2": [ "58", "376" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose the scores of students on an exam are Normally distributed with a mean of 291 and a standard deviation of 72. Then approximately 99.7\\% of the exam scores lie between the numbers [ANS] and [ANS] such that the mean is halfway between these two integers.", "answer_v3": [ "75", "507" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0262", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "A random variable $X$ is normally distributed, with a mean of 35 and a standard deviation of 3.8. Which of the following is the appropriate interquartile range for this distribution? [ANS] A. $44.75-25.25=19.50$ B. $35.95-34.05=1.90$ C. $37.56-32.44=5.12$ D. $39.69-30.31=9.38$ E. $36.23-33.77=2.46$", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "A random variable $X$ is normally distributed, with a mean of 21 and a standard deviation of 5.6. Which of the following is the appropriate interquartile range for this distribution? [ANS] A. $22.40-19.60=2.80$ B. $22.82-19.18=3.64$ C. $31.19-10.81=20.38$ D. $24.78-17.22=7.56$ E. $42.17+0.17=42.34$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "A random variable $X$ is normally distributed, with a mean of 26 and a standard deviation of 4. Which of the following is the appropriate interquartile range for this distribution? [ANS] A. $27.00-25.00=2.00$ B. $28.70-23.30=5.40$ C. $36.80-15.20=21.60$ D. $31.20-20.80=10.40$ E. $27.30-24.70=2.60$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0263", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "Given that $Z$ is a standard normal random variable, the value $z$ for which $P(Z \\leq z)=0.2580$ is [ANS] A.-0.6495 B. 0.6999 C. 0.3982 D. 0.6018\nIf a random variable $X$ is exponentially distributed with parameter $\\lambda=2$, then $P(X \\geq 1)$ is equal to [ANS] A. 0.6065 B. 0.8647 C. 0.1353 D. 0.3935", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Given that $Z$ is a standard normal random variable, the value $z$ for which $P(Z \\leq z)=0.2580$ is [ANS] A. 0.3982 B. 0.6999 C. 0.6018 D.-0.6495\nGiven that $Z$ is a standard normal random variable, $P(-1.0 \\leq Z \\leq 1.5)$ is equal to [ANS] A. 0.7745 B. 0.9332 C. 0.8413 D. 0.0919", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Given that $Z$ is a standard normal random variable, the value $z$ for which $P(Z \\leq z)=0.2580$ is [ANS] A. 0.3982 B.-0.6495 C. 0.6999 D. 0.6018\nIf $X$ is a normal random variable with a standard deviation of 10, then $3X$ has a standard deviation equal to [ANS] A. 30 B. 10 C. 90 D. $\\sqrt{30}$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0264", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. Using the standard normal curve, the $z-$ score representing the 75th percentile is 0.67. [ANS] 2. Using the standard normal curve, the $z-$ score representing the 90th percentile is 1.28. [ANS] 3. The mean and standard deviation of an exponential random variable are equal to each other. [ANS] 4. A random variable $X$ is normally distributed with a mean of 150 and a variance of 36. Given that $X=120$, its corresponding $z-$ score is 5.0", "answer_v1": [ "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. Using the standard normal curve, the $z-$ score representing the 90th percentile is 1.28. [ANS] 2. A random variable $X$ is normally distributed with a mean of 150 and a variance of 36. Given that $X=120$, its corresponding $z-$ score is 5.0 [ANS] 3. Using the standard normal curve, the $z-$ score representing the 75th percentile is 0.67. [ANS] 4. Let $z_1$ be a $z-$ score that is unknown but identifiable by position and area. If the symmetrical area between $-z_1$ and $+z_1$ is 0.9544, the value of $z_1$ is 2.0", "answer_v2": [ "T", "F", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. Let $z_1$ be a $z-$ score that is unknown but identifiable by position and area. If the area to the right of $z_1$ is 0.8413, the value of $z_1$ is 1.0 [ANS] 2. Let $z_1$ be a $z-$ score that is unknown but identifiable by position and area. If the symmetrical area between $-z_1$ and $+z_1$ is 0.9544, the value of $z_1$ is 2.0 [ANS] 3. Using the standard normal curve, the $z-$ score representing the 75th percentile is 0.67. [ANS] 4. The mean and standard deviation of a normally distributed random variable which has been standardized are one and zero, respectively.", "answer_v3": [ "F", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Probability_0265", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "Suppose that $X$ is normally distributed with mean 115 and standard deviation 22. A. What is the probability that $X$ is greater than 152.4? Probability=[ANS]\nB. What value of $X$ does only the top 17\\% exceed? $X$=[ANS]", "answer_v1": [ "0.0445654617719571", "135.99163" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that $X$ is normally distributed with mean 75 and standard deviation 29. A. What is the probability that $X$ is greater than 113.28? Probability=[ANS]\nB. What value of $X$ does only the top 13\\% exceed? $X$=[ANS]", "answer_v2": [ "0.0934175080068911", "107.66531" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that $X$ is normally distributed with mean 90 and standard deviation 22. A. What is the probability that $X$ is greater than 121.24? Probability=[ANS]\nB. What value of $X$ does only the top 16\\% exceed? $X$=[ANS]", "answer_v3": [ "0.0778038395399642", "111.878076" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0266", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "Compute the following probabilities for the standard normal distribution $Z$. A. $P(0 < Z < 2.3)$=[ANS]\nB. $P(-1.4 < Z < 0.6)$=[ANS]\nC. $P(Z >-1.25)$=[ANS]", "answer_v1": [ "0.489275889978324", "0.644990223016155", "0.894350225348758" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Compute the following probabilities for the standard normal distribution $Z$. A. $P(0 < Z < 1.85)$=[ANS]\nB. $P(-1.05 < Z < 0.2)$=[ANS]\nC. $P(Z >-1.65)$=[ANS]", "answer_v2": [ "0.467843225204386", "0.432400653063207", "0.950528530969186" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Compute the following probabilities for the standard normal distribution $Z$. A. $P(0 < Z < 2)$=[ANS]\nB. $P(-1.4 < Z < 0.3)$=[ANS]\nC. $P(Z >-1.45)$=[ANS]", "answer_v3": [ "0.477249868051821", "0.537154762955181", "0.926470739401079" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0267", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "A standard normal distribution is a normal distribution with [ANS] A. a mean of zero and a standard deviation of one B. a mean always larger than the standard deviation C. a mean usually larger than the standard deviation D. a mean of one and a standard deviation of zero\nWhich of the following is always true for all probability density functions of continuous random variables? [ANS] A. They have the same height B. They are symmetrical C. The area under the curve is 1.0 D. They are bell-shaped", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "A standard normal distribution is a normal distribution with [ANS] A. a mean usually larger than the standard deviation B. a mean always larger than the standard deviation C. a mean of one and a standard deviation of zero D. a mean of zero and a standard deviation of one\nGiven that $X$ is a normal variable, which of the following statements is true? [ANS] A. The variable $5X$ is also normally distributed B. The variable $X+5$ is also normally distributed C. The variable $X-5$ is also normally distributed D. All of the above statements is true", "answer_v2": [ "D", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "A standard normal distribution is a normal distribution with [ANS] A. a mean usually larger than the standard deviation B. a mean of zero and a standard deviation of one C. a mean always larger than the standard deviation D. a mean of one and a standard deviation of zero\nLike the normal distribution, the exponential density function $f(x)$ [ANS] A. approaches zero as $x$ approaches infinity B. is symmetrical C. is bell-shaped D. approaches infinity as $x$ approaches zero", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0268", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "Given a normal population whose mean is 600 and whose standard deviation is 55, find each of the following: A. The probability that a random sample of 6 has a mean between 605 and 621. Probability=[ANS]\nB. The probability that a random sample of 18 has a mean between 605 and 621. Probability=[ANS]\nC. The probability that a random sample of 23 has a mean between 605 and 621. Probability=[ANS]", "answer_v1": [ "0.23706470527587", "0.297236412538259", "0.297883773391102" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Given a normal population whose mean is 330 and whose standard deviation is 76, find each of the following: A. The probability that a random sample of 3 has a mean between 336 and 352. Probability=[ANS]\nB. The probability that a random sample of 15 has a mean between 336 and 352. Probability=[ANS]\nC. The probability that a random sample of 30 has a mean between 336 and 352. Probability=[ANS]", "answer_v2": [ "0.137567062670096", "0.248776036218789", "0.276295850185107" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Given a normal population whose mean is 425 and whose standard deviation is 56, find each of the following: A. The probability that a random sample of 4 has a mean between 432 and 450. Probability=[ANS]\nB. The probability that a random sample of 16 has a mean between 432 and 450. Probability=[ANS]\nC. The probability that a random sample of 22 has a mean between 432 and 450. Probability=[ANS]", "answer_v3": [ "0.215326834069988", "0.271464773170283", "0.260703445305469" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0269", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "2", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "A sample of $n=22$ observations is drawn from a normal population with $\\mu=1020$ and $\\sigma=210$. Find each of the following: A. $P(\\bar{X} > 1109)$ Probability=[ANS]\nB. $P(\\bar{X} < 943)$ Probability=[ANS]\nC. $P(\\bar{X} > 997)$ Probability=[ANS]", "answer_v1": [ "0.0234145344213119", "0.0427326627321883", "0.696273389286652" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A sample of $n=11$ observations is drawn from a normal population with $\\mu=1090$ and $\\sigma=160$. Find each of the following: A. $P(\\bar{X} > 1172)$ Probability=[ANS]\nB. $P(\\bar{X} < 983)$ Probability=[ANS]\nC. $P(\\bar{X} > 1065)$ Probability=[ANS]", "answer_v2": [ "0.0445870779960577", "0.0132776597109504", "0.697848523064152" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A sample of $n=15$ observations is drawn from a normal population with $\\mu=1020$ and $\\sigma=180$. Find each of the following: A. $P(\\bar{X} > 1108)$ Probability=[ANS]\nB. $P(\\bar{X} < 945)$ Probability=[ANS]\nC. $P(\\bar{X} > 996)$ Probability=[ANS]", "answer_v3": [ "0.0291484584206642", "0.0532915838008587", "0.697211690847254" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0270", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Gaussian normal", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "Independent random samples of $n=18$ observations each are drawn from normal populations. The parameters of these populations are:\nPopulation 1: $\\mu=278$ and $\\sigma=25$ Population 2: $\\mu=268$ and $\\sigma=29$\nFind the probability that the mean of sample 1 is greater than the mean of sample 2 by more than 16. Probability=[ANS]", "answer_v1": [ "0.253074763267878" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Independent random samples of $n=10$ observations each are drawn from normal populations. The parameters of these populations are:\nPopulation 1: $\\mu=280$ and $\\sigma=25$ Population 2: $\\mu=266$ and $\\sigma=29$\nFind the probability that the mean of sample 1 is greater than the mean of sample 2 by more than 16. Probability=[ANS]", "answer_v2": [ "0.434400321838692" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Independent random samples of $n=13$ observations each are drawn from normal populations. The parameters of these populations are:\nPopulation 1: $\\mu=278$ and $\\sigma=25$ Population 2: $\\mu=267$ and $\\sigma=29$\nFind the probability that the mean of sample 1 is greater than the mean of sample 2 by more than 17. Probability=[ANS]", "answer_v3": [ "0.286033565623329" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0271", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "Joint Distribution", "Normal Distribution", "Mean", "Standard Deviation" ], "problem_v1": "Andrew's bowling scores are approximately normally distributed with mean $140$ and standard deviation $21,$ while Jill's scores are normally distributed with mean $125$ and standard deviation $14.$ If Andrew and Jill each bowl one game, then assuming that their scores are independent random variables, approximate the probability that the total of their scores is above $270.$ [ANS]", "answer_v1": [ "0.421481" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Ted's bowling scores are approximately normally distributed with mean $110$ and standard deviation $15,$ while Leo's scores are normally distributed with mean $165$ and standard deviation $14.$ If Ted and Leo each bowl one game, then assuming that their scores are independent random variables, approximate the probability that the total of their scores is above $260.$ [ANS]", "answer_v2": [ "0.767627" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Mike's bowling scores are approximately normally distributed with mean $120$ and standard deviation $19,$ while Pam's scores are normally distributed with mean $115$ and standard deviation $14.$ If Mike and Pam each bowl one game, then assuming that their scores are independent random variables, approximate the probability that the total of their scores is above $250.$ [ANS]", "answer_v3": [ "0.262528" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0272", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "Probability", "Central Limit Theorem", "Normal", "Mean", "Standard Deviation" ], "problem_v1": "Scores for men on the verbal portion of the SAT-I test are normally distributed with a mean of $509$ and a standard deviation of $112$.\n(a) If $1$ man is randomly selected, find the probability that his score is at least $581$. [ANS]\n(b) If $18$ men are randomly selected, find the probability that their mean score is at least $581$. [ANS]\n$18$ randomly selected men were given a review course before taking the SAT test. If their mean score is $581$, is there a strong evidence to support the claim that the course is actually effective? (Enter YES or NO) $\\ $ [ANS]", "answer_v1": [ "0.260158399267029", "0.00319166434710809", "YES" ], "answer_type_v1": [ "NV", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "Scores for men on the verbal portion of the SAT-I test are normally distributed with a mean of $509$ and a standard deviation of $112$.\n(a) If $1$ man is randomly selected, find the probability that his score is at least $588$. [ANS]\n(b) If $10$ men are randomly selected, find the probability that their mean score is at least $588$. [ANS]\n$10$ randomly selected men were given a review course before taking the SAT test. If their mean score is $588$, is there a strong evidence to support the claim that the course is actually effective? (Enter YES or NO) $\\ $ [ANS]", "answer_v2": [ "0.240294002848723", "0.0128559678089529", "YES" ], "answer_type_v2": [ "NV", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "Scores for men on the verbal portion of the SAT-I test are normally distributed with a mean of $509$ and a standard deviation of $112$.\n(a) If $1$ man is randomly selected, find the probability that his score is at least $581.5$. [ANS]\n(b) If $13$ men are randomly selected, find the probability that their mean score is at least $581.5$. [ANS]\n$13$ randomly selected men were given a review course before taking the SAT test. If their mean score is $581.5$, is there a strong evidence to support the claim that the course is actually effective? (Enter YES or NO) $\\ $ [ANS]", "answer_v3": [ "0.258711966859857", "0.0097991533322115", "YES" ], "answer_type_v3": [ "NV", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0273", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "Probability", "Central Limit Theorem", "Normal", "Mean", "Standard Deviation" ], "problem_v1": "Assume that women's weights are normally distributed with a mean given by $\\mu=143$ lb and a standard deviation given by $\\sigma=29$ lb.\n(a) If 1 woman is randomly selected, find the probabity that her weight is between $113$ lb and $177$ lb [ANS]\n(b) If $6$ women are randomly selected, find the probability that they have a mean weight between $113$ lb and $177$ lb [ANS]\n(c) If $79$ women are randomly selected, find the probability that they have a mean weight between $113$ lb and $177$ lb [ANS]", "answer_v1": [ "0.729029251322942", "0.992320232351307", "1.00000000082191" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Assume that women's weights are normally distributed with a mean given by $\\mu=143$ lb and a standard deviation given by $\\sigma=29$ lb.\n(a) If 1 woman is randomly selected, find the probabity that her weight is below $108$ [ANS]\n(b) If $3$ women are randomly selected, find the probability that they have a mean weight below $108$ [ANS]\n(c) If $97$ women are randomly selected, find the probability that they have a mean weight below $108$ [ANS]", "answer_v2": [ "0.113735991933482", "0.0182906646027234", "-9.86587649471722E-10" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Assume that women's weights are normally distributed with a mean given by $\\mu=143$ lb and a standard deviation given by $\\sigma=29$ lb.\n(a) If 1 woman is randomly selected, find the probabity that her weight is between $108$ lb and $176$ lb [ANS]\n(b) If $4$ women are randomly selected, find the probability that they have a mean weight between $108$ lb and $176$ lb [ANS]\n(c) If $80$ women are randomly selected, find the probability that they have a mean weight between $108$ lb and $176$ lb [ANS]", "answer_v3": [ "0.758689366529648", "0.980679154699116", "1.0000001642573" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0274", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "Probability", "Central Limit Theorem", "Mean", "Standard Deviation" ], "problem_v1": "A soft drink bottler purchases glass bottles from a vendor. The bottles are required to have an internal pressure of at least $150$ pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with a mean internal pressure of $157$ psi and a standard deviation of $3$ psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the production process to verify the claim. the bottler randomly selects $70$ bottles from the last $10000$ produced, measures the internal pressure of each, and finds the mean pressure for the sample to be $0.6$ psi below the process mean cited by the vendor.\n(a) Assuming that the vendor is correct in his claim, what is the probability of obtaining a sample mean this far or farther below the process mean? [ANS]\n(b) If the standard deviation were $3$ psi as claimed, but the mean was $156$ psi, what is the probability of obtaining a sample mean of $156.4$ psi or below? [ANS] (c) If the process mean were $157$ psi as claimed, but the standard deviation was $3.2$ psi, what is the probability of obtaining a sample mean of $156.4$ psi or below? [ANS]", "answer_v1": [ "0.047132152434021", "0.867691890475346", "0.0583545516465445" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A soft drink bottler purchases glass bottles from a vendor. The bottles are required to have an internal pressure of at least $150$ pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with a mean internal pressure of $157$ psi and a standard deviation of $3$ psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the production process to verify the claim. the bottler randomly selects $40$ bottles from the last $10000$ produced, measures the internal pressure of each, and finds the mean pressure for the sample to be $1$ psi below the process mean cited by the vendor.\n(a) Assuming that the vendor is correct in his claim, what is the probability of obtaining a sample mean this far or farther below the process mean? [ANS]\n(b) If the standard deviation were $3$ psi as claimed, but the mean was $151$ psi, what is the probability of obtaining a sample mean of $156$ psi or below? [ANS] (c) If the process mean were $157$ psi as claimed, but the standard deviation was $2$ psi, what is the probability of obtaining a sample mean of $156$ psi or below? [ANS]", "answer_v2": [ "0.0175074895232436", "1.00000003235469", "0.00078270014241363" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A soft drink bottler purchases glass bottles from a vendor. The bottles are required to have an internal pressure of at least $150$ pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with a mean internal pressure of $157$ psi and a standard deviation of $3$ psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the production process to verify the claim. the bottler randomly selects $50$ bottles from the last $10000$ produced, measures the internal pressure of each, and finds the mean pressure for the sample to be $0.7$ psi below the process mean cited by the vendor.\n(a) Assuming that the vendor is correct in his claim, what is the probability of obtaining a sample mean this far or farther below the process mean? [ANS]\n(b) If the standard deviation were $3$ psi as claimed, but the mean was $153$ psi, what is the probability of obtaining a sample mean of $156.3$ psi or below? [ANS] (c) If the process mean were $157$ psi as claimed, but the standard deviation was $2.7$ psi, what is the probability of obtaining a sample mean of $156.3$ psi or below? [ANS]", "answer_v3": [ "0.0494800760231202", "0.999999993934334", "0.0333834575363845" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0275", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation" ], "problem_v1": "The physical fitness of an athlete is often measured by how much oxygen the athlete takes in (which is recorded in milliliters per kilogram, ml/kg). The mean maximum oxygen uptake for elite athletes has been found to be $75$ with a standard deviation of $7.9$. Assume that the distribution is approximately normal.\n(a) What is the probability that an elite athlete has a maximum oxygen uptake of at least $65$ ml/kg? answer: [ANS]\n(b) What is the probability that an elite athlete has a maximum oxygen uptake of $70$ ml/kg or lower? answer: [ANS]\n(c) Consider someone with a maximum oxygen uptake of $26$ ml/kg. Is it likely that this person is an elite athlete? Write \"YES\" or \"NO.\" answer: [ANS]", "answer_v1": [ "0.897211731131124", "0.263395752920759", "NO" ], "answer_type_v1": [ "NV", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "The physical fitness of an athlete is often measured by how much oxygen the athlete takes in (which is recorded in milliliters per kilogram, ml/kg). The mean maximum oxygen uptake for elite athletes has been found to be $60$ with a standard deviation of $9.7$. Assume that the distribution is approximately normal.\n(a) What is the probability that an elite athlete has a maximum oxygen uptake of at least $50$ ml/kg? answer: [ANS]\n(b) What is the probability that an elite athlete has a maximum oxygen uptake of $55$ ml/kg or lower? answer: [ANS]\n(c) Consider someone with a maximum oxygen uptake of $39$ ml/kg. Is it likely that this person is an elite athlete? Write \"YES\" or \"NO.\" answer: [ANS]", "answer_v2": [ "0.848712667827444", "0.303114437731165", "NO" ], "answer_type_v2": [ "NV", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "The physical fitness of an athlete is often measured by how much oxygen the athlete takes in (which is recorded in milliliters per kilogram, ml/kg). The mean maximum oxygen uptake for elite athletes has been found to be $65$ with a standard deviation of $8$. Assume that the distribution is approximately normal.\n(a) What is the probability that an elite athlete has a maximum oxygen uptake of at least $55$ ml/kg? answer: [ANS]\n(b) What is the probability that an elite athlete has a maximum oxygen uptake of $60$ ml/kg or lower? answer: [ANS]\n(c) Consider someone with a maximum oxygen uptake of $24$ ml/kg. Is it likely that this person is an elite athlete? Write \"YES\" or \"NO.\" answer: [ANS]", "answer_v3": [ "0.894350225348758", "0.26598552806204", "NO" ], "answer_type_v3": [ "NV", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0276", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation" ], "problem_v1": "Healty people have body temperatures that are normally distributed with a mean of $98.20^{\\circ}F$ and a standard deviation of $0.62^{\\circ}F$.\n(a) If a healthy person is randomly selected, what is the probability that he or she has a temperature above $99.6^{\\circ}F$? answer: [ANS]\n(b) A hospital wants to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 1.5 \\% of healty people to exceed it? answer: [ANS]", "answer_v1": [ "0.011970817751345", "99.5454560344227" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Healty people have body temperatures that are normally distributed with a mean of $98.20^{\\circ}F$ and a standard deviation of $0.62^{\\circ}F$.\n(a) If a healthy person is randomly selected, what is the probability that he or she has a temperature above $98.4^{\\circ}F$? answer: [ANS]\n(b) A hospital wants to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 2.5 \\% of healty people to exceed it? answer: [ANS]", "answer_v2": [ "0.373506425848973", "99.4151776704868" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Healty people have body temperatures that are normally distributed with a mean of $98.20^{\\circ}F$ and a standard deviation of $0.62^{\\circ}F$.\n(a) If a healthy person is randomly selected, what is the probability that he or she has a temperature above $98.8^{\\circ}F$? answer: [ANS]\n(b) A hospital wants to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 2 \\% of healty people to exceed it? answer: [ANS]", "answer_v3": [ "0.16658663349289", "99.4733243247358" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0277", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation" ], "problem_v1": "Using diaries for many weeks, a study on the lifestyles of visually impaired students was conducted. The students kept track of many lifestyle variables including how many hours of sleep obtained on a typical day. Researchers found that visually impaired students averaged $9.51$ hours of sleep, with a standard deviation of $2.16$ hours. Assume that the number of hours of sleep for these visually impaired students is normally distributed.\n(a) What is the probability that a visually impaired student gets less than $6.6$ hours of sleep? answer: [ANS]\n(b) What is the probability that a visually impaired student gets between $6.7$ and $8.55$ hours of sleep? answer: [ANS]\n(c) Twenty percent of students get less than how many hours of sleep on a typical day? answer: [ANS] hours", "answer_v1": [ "0.0889543354938676", "0.23171873783558", "7.69209813548251" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Using diaries for many weeks, a study on the lifestyles of visually impaired students was conducted. The students kept track of many lifestyle variables including how many hours of sleep obtained on a typical day. Researchers found that visually impaired students averaged $8.16$ hours of sleep, with a standard deviation of $2.87$ hours. Assume that the number of hours of sleep for these visually impaired students is normally distributed.\n(a) What is the probability that a visually impaired student gets less than $6.1$ hours of sleep? answer: [ANS]\n(b) What is the probability that a visually impaired student gets between $6.3$ and $10.82$ hours of sleep? answer: [ANS]\n(c) Twenty percent of students get less than how many hours of sleep on a typical day? answer: [ANS] hours", "answer_v2": [ "0.236449545214015", "0.564526974293147", "5.74454705964574" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Using diaries for many weeks, a study on the lifestyles of visually impaired students was conducted. The students kept track of many lifestyle variables including how many hours of sleep obtained on a typical day. Researchers found that visually impaired students averaged $8.63$ hours of sleep, with a standard deviation of $2.21$ hours. Assume that the number of hours of sleep for these visually impaired students is normally distributed.\n(a) What is the probability that a visually impaired student gets less than $6.3$ hours of sleep? answer: [ANS]\n(b) What is the probability that a visually impaired student gets between $6.6$ and $8.22$ hours of sleep? answer: [ANS]\n(c) Thirty percent of students get less than how many hours of sleep on a typical day? answer: [ANS] hours", "answer_v3": [ "0.14587310403782", "0.247245490761325", "7.47107486691523" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0278", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation", "Percentile" ], "problem_v1": "Assume that the readings on the thermometers are normally idstributed with a mean of $0^{\\circ}$ and a standard deviation of $1.00^{\\circ} \\mbox{C}$. Find $P_{75}$, the $75^{th}$ percentile. This is the temperature reading separating the bottom 75 \\% from the top 25 \\%. [ANS]", "answer_v1": [ "0.674489750196071" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Assume that the readings on the thermometers are normally idstributed with a mean of $0^{\\circ}$ and a standard deviation of $1.00^{\\circ} \\mbox{C}$. Find $P_{10}$, the $10^{th}$ percentile. This is the temperature reading separating the bottom 10 \\% from the top 90 \\%. [ANS]", "answer_v2": [ "-1.28155156554455" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Assume that the readings on the thermometers are normally idstributed with a mean of $0^{\\circ}$ and a standard deviation of $1.00^{\\circ} \\mbox{C}$. Find $P_{30}$, the $30^{th}$ percentile. This is the temperature reading separating the bottom 30 \\% from the top 70 \\%. [ANS]", "answer_v3": [ "-0.52440051270804" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0279", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation" ], "problem_v1": "Suppose that the readings on the thermometers are normally distributed with a mean of $0^{\\circ}$ and a standard deviation of $1.00^{\\circ} C$. If 10\\% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the reading that separates the rejected thermometers from the others. [ANS]", "answer_v1": [ "1.28155156554455" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the readings on the thermometers are normally distributed with a mean of $0^{\\circ}$ and a standard deviation of $1.00^{\\circ} C$. If 4\\% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the reading that separates the rejected thermometers from the others. [ANS]", "answer_v2": [ "1.75068607127288" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the readings on the thermometers are normally distributed with a mean of $0^{\\circ}$ and a standard deviation of $1.00^{\\circ} C$. If 6\\% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the reading that separates the rejected thermometers from the others. [ANS]", "answer_v3": [ "1.55477359460009" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0280", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation" ], "problem_v1": "The combined math and verbal scores for females taking the SAT-I test are normally distributed with a mean of 998 and a standard deviation of 202 (based on date from the College Board). If a college includes a minimum score of 1025 among its requirements, what percentage of females do not satisfy that requirement? [ANS]", "answer_v1": [ "55.3165611720739" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The combined math and verbal scores for females taking the SAT-I test are normally distributed with a mean of 998 and a standard deviation of 202 (based on date from the College Board). If a college includes a minimum score of 825 among its requirements, what percentage of females do not satisfy that requirement? [ANS]", "answer_v2": [ "19.587842709373" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The combined math and verbal scores for females taking the SAT-I test are normally distributed with a mean of 998 and a standard deviation of 202 (based on date from the College Board). If a college includes a minimum score of 900 among its requirements, what percentage of females do not satisfy that requirement? [ANS]", "answer_v3": [ "31.3785499037692" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0281", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation", "Probability" ], "problem_v1": "The extract of a plant native to Taiwan has been tested as a possible treatment for Leukemia. One of the chemical compounds produced from the plant was analyzed for a particular collagen. The collagen amount was found to be normally distributed with a mean of $75$ and standard deviation of $7.9$ grams per mililiter.\n(a) What is the probability that the amount of collagen is greater than $69$ grams per mililiter? answer: [ANS]\n(b) What is the probability that the amount of collagen is less than $85$ grams per mililiter? answer: [ANS]\n(c) What percentage of compounds formed from the extract of this plant fall within $1$ standard deviations of the mean? answer: [ANS] \\%", "answer_v1": [ "0.776221349665108", "0.897211731131124", "68.2689492137085" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The extract of a plant native to Taiwan has been tested as a possible treatment for Leukemia. One of the chemical compounds produced from the plant was analyzed for a particular collagen. The collagen amount was found to be normally distributed with a mean of $61$ and standard deviation of $9.7$ grams per mililiter.\n(a) What is the probability that the amount of collagen is greater than $60$ grams per mililiter? answer: [ANS]\n(b) What is the probability that the amount of collagen is less than $77$ grams per mililiter? answer: [ANS]\n(c) What percentage of compounds formed from the extract of this plant fall within $3$ standard deviations of the mean? answer: [ANS] \\%", "answer_v2": [ "0.54105533279559", "0.950475794669562", "99.7300203936223" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The extract of a plant native to Taiwan has been tested as a possible treatment for Leukemia. One of the chemical compounds produced from the plant was analyzed for a particular collagen. The collagen amount was found to be normally distributed with a mean of $66$ and standard deviation of $8$ grams per mililiter.\n(a) What is the probability that the amount of collagen is greater than $61$ grams per mililiter? answer: [ANS]\n(b) What is the probability that the amount of collagen is less than $81$ grams per mililiter? answer: [ANS]\n(c) What percentage of compounds formed from the extract of this plant fall within $1$ standard deviations of the mean? answer: [ANS] \\%", "answer_v3": [ "0.734014469966969", "0.969603637227035", "68.2689492137085" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0282", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "Normal Distribution", "Mensa", "Standard Deviation" ], "problem_v1": "IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. is an international society that has one-and only one-qualification for membership: a score in the top 2\\% of the population on an IQ test.\n(a) What IQ score should one have in order to be eligible for Mensa? [ANS]\n(b) In a typical region of 155,000 people, how many are eligible for Mensa? [ANS]", "answer_v1": [ "130.81", "3100" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. is an international society that has one-and only one-qualification for membership: a score in the top 2\\% of the population on an IQ test.\n(a) What IQ score should one have in order to be eligible for Mensa? [ANS]\n(b) In a typical region of 35,000 people, how many are eligible for Mensa? [ANS]", "answer_v2": [ "130.81", "700" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. is an international society that has one-and only one-qualification for membership: a score in the top 2\\% of the population on an IQ test.\n(a) What IQ score should one have in order to be eligible for Mensa? [ANS]\n(b) In a typical region of 75,000 people, how many are eligible for Mensa? [ANS]", "answer_v3": [ "130.81", "1500" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0283", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "Normal Distribution", "Mean", "Standard Deviation", "Decile" ], "problem_v1": "Women's weights are normally distributed with a mean given by $\\mu=143$ lb and a standard deviation given by $\\sigma=29$ lb. Find the seventh decile, $D_{7}$, which separates the bottom 70\\% from the top 30\\%. [ANS]", "answer_v1": [ "158.207614868533" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Women's weights are normally distributed with a mean given by $\\mu=143$ lb and a standard deviation given by $\\sigma=29$ lb. Find the first decile, $D_{1}$, which separates the bottom 10\\% from the top 90\\%. [ANS]", "answer_v2": [ "105.835004599208" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Women's weights are normally distributed with a mean given by $\\mu=143$ lb and a standard deviation given by $\\sigma=29$ lb. Find the third decile, $D_{3}$, which separates the bottom 30\\% from the top 70\\%. [ANS]", "answer_v3": [ "127.792385131467" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0284", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "An automobile insurer has found that repair claims are Normally distributed with a mean of \\$840 and a standard deviation of \\$790.\n(a) Find the probability that a single claim, chosen at random, will be less than \\$800. ANSWER: [ANS]\n(b) Now suppose that the next 80 claims can be regarded as a random sample from the long-run claims process. Find the probability that the average $\\bar x$ of the 80 claims is smaller than \\$800. ANSWER: [ANS]\n(c) If a sample larger than 80 claims is considered, there would be [ANS] chance of getting a sample with an average smaller then \\$800. (NOTE: Enter ''LESS'', ''MORE'' or ''AN EQUAL'' without the quotes.)", "answer_v1": [ "0.480061", "0.326355", "LESS" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "LESS", "MORE", "AN EQUAL" ] ], "problem_v2": "An automobile insurer has found that repair claims are Normally distributed with a mean of \\$530 and a standard deviation of \\$470.\n(a) Find the probability that a single claim, chosen at random, will be less than \\$500. ANSWER: [ANS]\n(b) Now suppose that the next 50 claims can be regarded as a random sample from the long-run claims process. Find the probability that the average $\\bar x$ of the 50 claims is smaller than \\$500. ANSWER: [ANS]\n(c) If a sample larger than 50 claims is considered, there would be [ANS] chance of getting a sample with an average smaller then \\$500. (NOTE: Enter ''LESS'', ''MORE'' or ''AN EQUAL'' without the quotes.)", "answer_v2": [ "0.476078", "0.326355", "LESS" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "LESS", "MORE", "AN EQUAL" ] ], "problem_v3": "An automobile insurer has found that repair claims are Normally distributed with a mean of \\$640 and a standard deviation of \\$590.\n(a) Find the probability that a single claim, chosen at random, will be less than \\$600. ANSWER: [ANS]\n(b) Now suppose that the next 60 claims can be regarded as a random sample from the long-run claims process. Find the probability that the average $\\bar x$ of the 60 claims is smaller than \\$600. ANSWER: [ANS]\n(c) If a sample larger than 60 claims is considered, there would be [ANS] chance of getting a sample with an average smaller then \\$600. (NOTE: Enter ''LESS'', ''MORE'' or ''AN EQUAL'' without the quotes.)", "answer_v3": [ "0.472097", "0.298056", "LESS" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "LESS", "MORE", "AN EQUAL" ] ] }, { "id": "Probability_0285", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "The distribution of actual weights of 8-oz chocolate bars produced by a certain machine is normal with mean 8.2 ounces and standard deviation 0.16 ounces.\n(a) What is the probability that the average weight of a bar in a Simple Random Sample (SRS) with four of these chocolate bars is between 8.03 and 8.33 ounces? ANSWER: [ANS]\n(b) For a SRS of four of these chocolate bars, what is the level $L$ such that there is a 2\\% chance that the average weight is less than $L$? ANSWER: [ANS]", "answer_v1": [ "0.931126", "8.0357" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The distribution of actual weights of 8-oz chocolate bars produced by a certain machine is normal with mean 7.7 ounces and standard deviation 0.2 ounces.\n(a) What is the probability that the average weight of a bar in a Simple Random Sample (SRS) with three of these chocolate bars is between 7.57 and 7.9 ounces? ANSWER: [ANS]\n(b) For a SRS of three of these chocolate bars, what is the level $L$ such that there is a 2\\% chance that the average weight is less than $L$? ANSWER: [ANS]", "answer_v2": [ "0.82825", "7.46285" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The distribution of actual weights of 8-oz chocolate bars produced by a certain machine is normal with mean 7.9 ounces and standard deviation 0.16 ounces.\n(a) What is the probability that the average weight of a bar in a Simple Random Sample (SRS) with three of these chocolate bars is between 7.74 and 8.02 ounces? ANSWER: [ANS]\n(b) For a SRS of three of these chocolate bars, what is the level $L$ such that there is a 2\\% chance that the average weight is less than $L$? ANSWER: [ANS]", "answer_v3": [ "0.861402", "7.71028" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0286", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "A company sells sunscreen in 450 milliliter (ml) tubes. In fact, the amount of lotion in a tube varies according to a normal distribution with mean $\\mu=447$ ml and standard deviation $\\sigma=5$ ml. Suppose a store which sells this sunscreen advertises a sale for 6 tubes for the price of 5. Consider the average amount of lotion from a SRS of 6 tubes of sunscreen and find:\n(a) The standard deviation of the average, $\\bar x$: [ANS]\n(b) The probability that the average amount of sunscreen from 6 tubes will be less than 441 ml. Answer: [ANS]", "answer_v1": [ "2.04124", "0.00164106" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A company sells sunscreen in 300 milliliter (ml) tubes. In fact, the amount of lotion in a tube varies according to a normal distribution with mean $\\mu=296$ ml and standard deviation $\\sigma=5$ ml. Suppose a store which sells this sunscreen advertises a sale for 5 tubes for the price of 4. Consider the average amount of lotion from a SRS of 5 tubes of sunscreen and find:\n(a) The standard deviation of the average, $\\bar x$: [ANS]\n(b) The probability that the average amount of sunscreen from 5 tubes will be less than 288 ml. Answer: [ANS]", "answer_v2": [ "2.23607", "0.000171797" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A company sells sunscreen in 350 milliliter (ml) tubes. In fact, the amount of lotion in a tube varies according to a normal distribution with mean $\\mu=347$ ml and standard deviation $\\sigma=4$ ml. Suppose a store which sells this sunscreen advertises a sale for 5 tubes for the price of 4. Consider the average amount of lotion from a SRS of 5 tubes of sunscreen and find:\n(a) The standard deviation of the average, $\\bar x$: [ANS]\n(b) The probability that the average amount of sunscreen from 5 tubes will be less than 342 ml. Answer: [ANS]", "answer_v3": [ "1.78885", "0.00255513" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0287", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistic", "parameter", "sampling distribution", "sample" ], "problem_v1": "Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 129 mg/dl and standard deviation 9 mg/dl. Let $L$ denote a patient's glucose level.\n(a) If measurements are made on four different days, find the level $L$ such that there is probability only 0.04 that the mean glucose level of four test results falls above $L$ for Shelia's glucose level distribution. What is the value of $L$? ANSWER: [ANS]\n(b) If the mean result from the four tests is compared to the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes? ANSWER: [ANS]", "answer_v1": [ "136.898", "0.00734363" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 127 mg/dl and standard deviation 10 mg/dl. Let $L$ denote a patient's glucose level.\n(a) If measurements are made on three different days, find the level $L$ such that there is probability only 0.02 that the mean glucose level of three test results falls above $L$ for Shelia's glucose level distribution. What is the value of $L$? ANSWER: [ANS]\n(b) If the mean result from the three tests is compared to the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes? ANSWER: [ANS]", "answer_v2": [ "138.882", "0.0122245" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 127 mg/dl and standard deviation 9 mg/dl. Let $L$ denote a patient's glucose level.\n(a) If measurements are made on three different days, find the level $L$ such that there is probability only 0.03 that the mean glucose level of three test results falls above $L$ for Shelia's glucose level distribution. What is the value of $L$? ANSWER: [ANS]\n(b) If the mean result from the three tests is compared to the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational diabetes? ANSWER: [ANS]", "answer_v3": [ "136.795", "0.00620967" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0288", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "The heights of women aged 20 to 29 are approximately Normal with mean 64 inches and standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches.\n(a) What is the z-score for a woman 72 inches tall? z-score=[ANS]\nWhat is the z-score for a man 74 inches tall? z-score=[ANS]", "answer_v1": [ "2.96296", "1.67857" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The heights of women aged 20 to 29 are approximately Normal with mean 64 inches and standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches.\n(a) What is the z-score for a woman 57 inches tall? z-score=[ANS]\nWhat is the z-score for a man 83 inches tall? z-score=[ANS]", "answer_v2": [ "-2.59259", "4.89286" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The heights of women aged 20 to 29 are approximately Normal with mean 64 inches and standard deviation 2.7 inches. Men the same age have mean height 69.3 inches with standard deviation 2.8 inches.\n(a) What is the z-score for a woman 62 inches tall? z-score=[ANS]\nWhat is the z-score for a man 75 inches tall? z-score=[ANS]", "answer_v3": [ "-0.740741", "2.03571" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0289", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "John took 4 courses last semester: History, Physics, Calculus, and Biology. The means and standard deviations for the final exams, and John's scores are given in the table below. Convert John's scores into z-scores.\n$\\begin{array}{ccccc}\\hline Subject & Mean & Stand. dev. & John's score & John's z-score \\\\ \\hline History & 53 & 16 & 61 & [ANS] \\\\ \\hline Physics & 60 & 14 & 60 & [ANS] \\\\ \\hline Calculus & 70 & 12 & 73 & [ANS] \\\\ \\hline Biology & 77 & 10 & 89.5 & [ANS] \\\\ \\hline \\end{array}$\nOn what exam did John have the highest relative score? (Enter the subject.) [ANS]", "answer_v1": [ "0.5", "0", "0.25", "1.25", "Biology" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [ "History", "Physics", "Calculus", "Biolog" ] ], "problem_v2": "Nick took 4 courses last semester: Calculus, Spanish, History, and Biology. The means and standard deviations for the final exams, and Nick's scores are given in the table below. Convert Nick's scores into z-scores.\n$\\begin{array}{ccccc}\\hline Subject & Mean & Stand. dev. & Nick's score & Nick's z-score \\\\ \\hline Calculus & 70 & 12 & 70 & [ANS] \\\\ \\hline Spanish & 44 & 12 & 56 & [ANS] \\\\ \\hline History & 53 & 16 & 37 & [ANS] \\\\ \\hline Biology & 77 & 10 & 89.5 & [ANS] \\\\ \\hline \\end{array}$\nOn what exam did Nick have the highest relative score? (Enter the subject.) [ANS]", "answer_v2": [ "0", "1", "-1", "1.25", "Biology" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [ "History", "Physics", "Calculus", "Biolog" ] ], "problem_v3": "Mary took 4 courses last semester: Biology, Spanish, Calculus, and Physics. The means and standard deviations for the final exams, and Mary's scores are given in the table below. Convert Mary's scores into z-scores.\n$\\begin{array}{ccccc}\\hline Subject & Mean & Stand. dev. & Mary's score & Mary's z-score \\\\ \\hline Biology & 77 & 10 & 94.5 & [ANS] \\\\ \\hline Spanish & 44 & 12 & 59 & [ANS] \\\\ \\hline Calculus & 70 & 12 & 64 & [ANS] \\\\ \\hline Physics & 60 & 14 & 56.5 & [ANS] \\\\ \\hline \\end{array}$\nOn what exam did Mary have the highest relative score? (Enter the subject.) [ANS]", "answer_v3": [ "1.75", "1.25", "-0.5", "-0.25", "Biology" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [ "History", "Physics", "Calculus", "Biolog" ] ] }, { "id": "Probability_0290", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "Scores on a certain intelligence test for children between ages 13 and 15 years are approximately normally distributed with $\\mu=111$ and $\\sigma=21$.\n(a) What proportion of children aged 13 to 15 years old have scores on this test above 97? (NOTE: Please enter your answer in decimal form. For example, 45.23\\% should be entered as 0.4523.) Answer: [ANS]\n(b) Enter the score which marks the lowest 30 percent of the distribution. Answer: [ANS]\n(c) Enter the score which marks the highest 5 percent of the distribution. Answer: [ANS]", "answer_v1": [ "0.747507", "99.9876", "145.542" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Scores on a certain intelligence test for children between ages 13 and 15 years are approximately normally distributed with $\\mu=96$ and $\\sigma=27$.\n(a) What proportion of children aged 13 to 15 years old have scores on this test above 86? (NOTE: Please enter your answer in decimal form. For example, 45.23\\% should be entered as 0.4523.) Answer: [ANS]\n(b) Enter the score which marks the lowest 25 percent of the distribution. Answer: [ANS]\n(c) Enter the score which marks the highest 15 percent of the distribution. Answer: [ANS]", "answer_v2": [ "0.644447", "77.7888", "123.984" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Scores on a certain intelligence test for children between ages 13 and 15 years are approximately normally distributed with $\\mu=101$ and $\\sigma=24$.\n(a) What proportion of children aged 13 to 15 years old have scores on this test above 91? (NOTE: Please enter your answer in decimal form. For example, 45.23\\% should be entered as 0.4523.) Answer: [ANS]\n(b) Enter the score which marks the lowest 25 percent of the distribution. Answer: [ANS]\n(c) Enter the score which marks the highest 5 percent of the distribution. Answer: [ANS]", "answer_v3": [ "0.661539", "84.8122", "140.476" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0291", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "1", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "IQ scores have a mean of 100 and a standard deviation of 15. John has an IQ of 124.\n(a) What is the difference between John's IQ and the mean? Answer: [ANS]\n(b) Convert John's IQ score to a z-score. Answer: [ANS]", "answer_v1": [ "24", "1.6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "IQ scores have a mean of 100 and a standard deviation of 15. Mary has an IQ of 103.\n(a) What is the difference between Mary's IQ and the mean? Answer: [ANS]\n(b) Convert Mary's IQ score to a z-score. Answer: [ANS]", "answer_v2": [ "3", "0.2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "IQ scores have a mean of 100 and a standard deviation of 15. Nick has an IQ of 112.\n(a) What is the difference between Nick's IQ and the mean? Answer: [ANS]\n(b) Convert Nick's IQ score to a z-score. Answer: [ANS]", "answer_v3": [ "12", "0.8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0292", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "Almost all medical schools in the United States require applicants to take the Medical College Admission Test (MCAT). On one exam, the scores of all applicants on the biological sciences part of the MCAT were approximately Normal with mean 9.7 and standard deviation 2.4. For applicants who actually entered medical school, the mean score was 10.6 and the standard deviation was 1.7.\n(a) What percent of all applicants had scores higher than 12? ANSWER: [ANS] \\% (b) What percent of those who entered medical school had scores between 8 and 11? ANSWER: [ANS] \\%", "answer_v1": [ "16.89", "52.99" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Almost all medical schools in the United States require applicants to take the Medical College Admission Test (MCAT). On one exam, the scores of all applicants on the biological sciences part of the MCAT were approximately Normal with mean 8.9 and standard deviation 2.6. For applicants who actually entered medical school, the mean score was 10.2 and the standard deviation was 1.5.\n(a) What percent of all applicants had scores higher than 12? ANSWER: [ANS] \\% (b) What percent of those who entered medical school had scores between 10 and 11? ANSWER: [ANS] \\%", "answer_v2": [ "11.66", "25.61" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Almost all medical schools in the United States require applicants to take the Medical College Admission Test (MCAT). On one exam, the scores of all applicants on the biological sciences part of the MCAT were approximately Normal with mean 9.2 and standard deviation 2.4. For applicants who actually entered medical school, the mean score was 10.3 and the standard deviation was 1.6.\n(a) What percent of all applicants had scores higher than 13? ANSWER: [ANS] \\% (b) What percent of those who entered medical school had scores between 8 and 11? ANSWER: [ANS] \\%", "answer_v3": [ "5.67", "59.38" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0293", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "The temperature at any random location in a kiln used in the manufacture of bricks is normally distributed with a mean of 1025 and a standard deviation of 50 degrees. If bricks are fired at a temperature above 1130, they will crack and must be disposed of. If the bricks are placed randomly throughout the kiln, the proportion of bricks that crack during the firing process is closest to [ANS]. (Note: Please enter your answer in decimal form. For example, 5.02\\% should be entered as 0.0502.)", "answer_v1": [ "0.0179" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The temperature at any random location in a kiln used in the manufacture of bricks is normally distributed with a mean of 900 and a standard deviation of 60 degrees. If bricks are fired at a temperature above 1095, they will crack and must be disposed of. If the bricks are placed randomly throughout the kiln, the proportion of bricks that crack during the firing process is closest to [ANS]. (Note: Please enter your answer in decimal form. For example, 5.02\\% should be entered as 0.0502.)", "answer_v2": [ "0.0006" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The temperature at any random location in a kiln used in the manufacture of bricks is normally distributed with a mean of 950 and a standard deviation of 50 degrees. If bricks are fired at a temperature above 1105, they will crack and must be disposed of. If the bricks are placed randomly throughout the kiln, the proportion of bricks that crack during the firing process is closest to [ANS]. (Note: Please enter your answer in decimal form. For example, 5.02\\% should be entered as 0.0502.)", "answer_v3": [ "0.001" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0294", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "For each problem, select the best response.\n(a) The scores of adults on an IQ test is approximately Normal with mean 100 and standard deviation 15. Corinne scores 118 on such test. Her z-score is about [ANS] A. 18 B. 1.2 C. 0.67 D. 7.87 E. None of the above.\n(b) To completely specify the shape of a Normal distribution, you must give [ANS] A. the median and the standard deviation. B. the five-number summary. C. the mean and the median. D. the mean and the standard deviation. E. All of the above.\n(c) The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. About 95\\% of all pregnancies last between [ANS] A. 250 and 282 days. B. 218 and 314 days. C. 260 and 320 days. D. 234 and 298 days. E. None of the above.", "answer_v1": [ "B", "D", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For each problem, select the best response.\n(a) Which of these variables is least likely to have a Normal distribution? [ANS] A. Heights of 100 white pine trees in a forest. B. Lengths of 50 newly hatched pythons. C. Income per person for 150 different countries. D. None of the above.\n(b) The scores of adults on an IQ test is approximately Normal with mean 100 and standard deviation 15. Corinne scores 118 on such test. Her z-score is about [ANS] A. 0.67 B. 1.2 C. 18 D. 7.87 E. None of the above.\n(c) To completely specify the shape of a Normal distribution, you must give [ANS] A. the mean and the standard deviation. B. the mean and the median. C. the median and the standard deviation. D. the five-number summary. E. All of the above.", "answer_v2": [ "C", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For each problem, select the best response.\n(a) To completely specify the shape of a Normal distribution, you must give [ANS] A. the mean and the standard deviation. B. the mean and the median. C. the five-number summary. D. the median and the standard deviation. E. All of the above.\n(b) The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. About 95\\% of all pregnancies last between [ANS] A. 218 and 314 days. B. 234 and 298 days. C. 260 and 320 days. D. 250 and 282 days. E. None of the above.\n(c) Which of these variables is least likely to have a Normal distribution? [ANS] A. Income per person for 150 different countries. B. Lengths of 50 newly hatched pythons. C. Heights of 100 white pine trees in a forest. D. None of the above.", "answer_v3": [ "A", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0295", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistic", "normal distribution", "z score" ], "problem_v1": "The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 rule to answer the questions in parts\n(a),(b),(c) below. Your answer must be entered in the correct format. If your answer is a percent, such as 25 percent, enter: \"25 PERCENT\" (without the quotes). If your answer is in inches, such as 10 inches, enter: \"10 INCHES\" (without the quotes and with a space between the number and the INCHES). If your answer is an interval, such as 14 to 15 inches, then enter: \"14 TO 15 INCHES\" (without the quotes). Don't report accuracy greater than what comes from the 68-95-99.7 rule! For example, if the rule says \"10 INCHES\" then \"10.0 INCHES\" is wrong because the extra decimal place claims too much accuracy. If the rule says \"1.2 INCHES\" then \"1.20 INCHES\" is wrong.\n(a) About what percent of men are between 64 and 66.5 inches? Answer: [ANS]\n(b) About what percent of men are taller than 69 inches? Answer: [ANS]\n(c) About what percent of men are between 69 and 74 inches? Answer: [ANS]", "answer_v1": [ "13.5", "50", "47.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 rule to answer the questions in parts\n(a),(b),(c) below. Your answer must be entered in the correct format. If your answer is a percent, such as 25 percent, enter: \"25 PERCENT\" (without the quotes). If your answer is in inches, such as 10 inches, enter: \"10 INCHES\" (without the quotes and with a space between the number and the INCHES). If your answer is an interval, such as 14 to 15 inches, then enter: \"14 TO 15 INCHES\" (without the quotes). Don't report accuracy greater than what comes from the 68-95-99.7 rule! For example, if the rule says \"10 INCHES\" then \"10.0 INCHES\" is wrong because the extra decimal place claims too much accuracy. If the rule says \"1.2 INCHES\" then \"1.20 INCHES\" is wrong.\n(a) About what percent of men are taller than 74? Answer: [ANS]\n(b) Fill in the blank: About 2.5 percent of all men are shorter than ________. Answer: [ANS]\n(c) Between what approximate heights do the middle 95 percent of men fall? Answer: [ANS]", "answer_v2": [ "2.5", "64 INCHES", "64 TO 74 INCHES" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 rule to answer the questions in parts\n(a),(b),(c) below. Your answer must be entered in the correct format. If your answer is a percent, such as 25 percent, enter: \"25 PERCENT\" (without the quotes). If your answer is in inches, such as 10 inches, enter: \"10 INCHES\" (without the quotes and with a space between the number and the INCHES). If your answer is an interval, such as 14 to 15 inches, then enter: \"14 TO 15 INCHES\" (without the quotes). Don't report accuracy greater than what comes from the 68-95-99.7 rule! For example, if the rule says \"10 INCHES\" then \"10.0 INCHES\" is wrong because the extra decimal place claims too much accuracy. If the rule says \"1.2 INCHES\" then \"1.20 INCHES\" is wrong.\n(a) About what percent of men are shorter than 66.5 inches? Answer: [ANS]\n(b) About what percent of men are between 69 and 74 inches? Answer: [ANS]\n(c) Between what approximate heights do the middle 95 percent of men fall? Answer: [ANS]", "answer_v3": [ "16", "47.5", "64 TO 74 INCHES" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0296", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "Probability", "Normal Distribution", "Central Limit Theorem", "Normal", "Mean", "Standard Deviation" ], "problem_v1": "Cans of regular Coke are labeled as containing $12 \\mbox{oz}$. Statistics students weighed the contents of $9$ randomly chosen cans, and found the mean weight to be $12.13$ ounces. Assume that cans of Coke are filled so that the actual amounts are normally distributed with a mean of $12.00 \\mbox{oz}$ and a standard deviation of $0.12 \\mbox{oz}$. Find the probability that a sample of $9$ cans will have a mean amount of at least $12.13 \\mbox{oz}$. [ANS]", "answer_v1": [ "0.000577024055803172" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Cans of regular Coke are labeled as containing $12 \\mbox{oz}$. Statistics students weighed the contents of $5$ randomly chosen cans, and found the mean weight to be $12.15$ ounces. Assume that cans of Coke are filled so that the actual amounts are normally distributed with a mean of $12.00 \\mbox{oz}$ and a standard deviation of $0.09 \\mbox{oz}$. Find the probability that a sample of $5$ cans will have a mean amount of at least $12.15 \\mbox{oz}$. [ANS]", "answer_v2": [ "9.69698279642112E-05" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Cans of regular Coke are labeled as containing $12 \\mbox{oz}$. Statistics students weighed the contents of $6$ randomly chosen cans, and found the mean weight to be $12.13$ ounces. Assume that cans of Coke are filled so that the actual amounts are normally distributed with a mean of $12.00 \\mbox{oz}$ and a standard deviation of $0.1 \\mbox{oz}$. Find the probability that a sample of $6$ cans will have a mean amount of at least $12.13 \\mbox{oz}$. [ANS]", "answer_v3": [ "0.000725429830092736" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0297", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [], "problem_v1": "Suppose the heights of men are normally distributed with mean, $\\mu$=69.5 inches, and standard deviation, $\\sigma$=4 inches. Suppose admission to a summer basketball camp requires that a camp participant must be in the top 30 \\% of men's heights, what is the minimum height that a camp participant can have in order to meet the camp's height admission requirement? answer: [ANS] inches", "answer_v1": [ "71.5976020508322" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose the heights of men are normally distributed with mean, $\\mu$=68 inches, and standard deviation, $\\sigma$=6 inches. Suppose admission to a summer basketball camp requires that a camp participant must be in the top 15 \\% of men's heights, what is the minimum height that a camp participant can have in order to meet the camp's height admission requirement? answer: [ANS] inches", "answer_v2": [ "74.2186003369628" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose the heights of men are normally distributed with mean, $\\mu$=68.5 inches, and standard deviation, $\\sigma$=5 inches. Suppose admission to a summer basketball camp requires that a camp participant must be in the top 15 \\% of men's heights, what is the minimum height that a camp participant can have in order to meet the camp's height admission requirement? answer: [ANS] inches", "answer_v3": [ "73.682166947469" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0298", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "empirical rule" ], "problem_v1": "The shelf life of a battery produced by one major company is known to be normally distributed, with a mean life of 4.3 years and a standard deviation of 1 years. Using the expanded empirical rule, what is the probability in decimal form that a randomly chosen battery will\n(a) last more than 1.3 years? Answer: [ANS]\n(b) last between 2.3 and 6.3 years? Answer: [ANS]\n(c) last fewer than 4.97 years? Answer: [ANS]", "answer_v1": [ "0.9985", "0.95", "0.75" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The shelf life of a battery produced by one major company is known to be normally distributed, with a mean life of 2.4 years and a standard deviation of 0.6 years. Using the expanded empirical rule, what is the probability in decimal form that a randomly chosen battery will\n(a) last between 0.6 and 4.2 years? Answer: [ANS]\n(b) last fewer than 3 years? Answer: [ANS]\n(c) last more than 1.998 years? Answer: [ANS]", "answer_v2": [ "0.997", "0.84", "0.75" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The shelf life of a battery produced by one major company is known to be normally distributed, with a mean life of 2.8 years and a standard deviation of 0.9 years. Using the expanded empirical rule, what is the probability in decimal form that a randomly chosen battery will\n(a) last between 2.197 and 3.403 years? Answer: [ANS]\n(b) last more than 4.6 years? Answer: [ANS]\n(c) last fewer than 5.5 years? Answer: [ANS]", "answer_v3": [ "0.5", "0.025", "0.9985" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0299", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "The weights of cans of Ocean brand tuna are supposed to have a net weight of 6 ounces. The manufacturer tells you that the net weight is actually a Normal random variable with a mean of 6.01 ounces and a standard deviation of 0.22 ounces. Suppose that you draw a random sample of 39 cans.\nPart i) Using the information about the distribution of the net weight given by the manufacturer, find the probability that the mean weight of the sample is less than 5.97 ounces. (Please carry answers to at least six decimal places in intermediate steps. Give your final answer to the nearest three decimal places). Probability (as a proportion) [ANS]\nPart ii) Use normal approximation to find the probability that more than 53.8\\% of the sampled cans are overweight (i.e. the net weight exceeds 6 ounces). [ANS] A. 0.5987 B. 0.5026 C. 0.4974 D. 0.4013", "answer_v1": [ "0.128092533550598", "D" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D" ] ], "problem_v2": "The weights of cans of Ocean brand tuna are supposed to have a net weight of 6 ounces. The manufacturer tells you that the net weight is actually a Normal random variable with a mean of 5.95 ounces and a standard deviation of 0.24 ounces. Suppose that you draw a random sample of 32 cans.\nPart i) Using the information about the distribution of the net weight given by the manufacturer, find the probability that the mean weight of the sample is less than 5.93 ounces. (Please carry answers to at least six decimal places in intermediate steps. Give your final answer to the nearest three decimal places). Probability (as a proportion) [ANS]\nPart ii) Use normal approximation to find the probability that more than 46.7\\% of the sampled cans are overweight (i.e. the net weight exceeds 6 ounces). [ANS] A. 0.7169 B. 0.2831 C. 0.4928 D. 0.5072", "answer_v2": [ "0.318675943130243", "B" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D" ] ], "problem_v3": "The weights of cans of Ocean brand tuna are supposed to have a net weight of 6 ounces. The manufacturer tells you that the net weight is actually a Normal random variable with a mean of 5.97 ounces and a standard deviation of 0.23 ounces. Suppose that you draw a random sample of 34 cans.\nPart i) Using the information about the distribution of the net weight given by the manufacturer, find the probability that the mean weight of the sample is less than 5.94 ounces. (Please carry answers to at least six decimal places in intermediate steps. Give your final answer to the nearest three decimal places). Probability (as a proportion) [ANS]\nPart ii) Use normal approximation to find the probability that more than 46.8\\% of the sampled cans are overweight (i.e. the net weight exceeds 6 ounces). [ANS] A. 0.5028 B. 0.4073 C. 0.4972 D. 0.5927", "answer_v3": [ "0.223460274355236", "B" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0300", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "continuous random variables", "probability", "uniform distribution" ], "problem_v1": "An exam consists of 46 multiple-choice questions. Each question has a choice of five answers, only one of which is correct. For each correct answer, a candidate gets 1 mark, and no penalty is applied for getting an incorrect answer. A particular candidate answers each question purely by guess-work. Using Normal approximation to Binomial distribution with continuity correction, what is the estimated probability this student obtains a score greater than or equal to 10? Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps. [ANS] A. 0.4560 B. 0.3159 C. 0.4567 D. 0.5440 E. 0.5163", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "An exam consists of 40 multiple-choice questions. Each question has a choice of five answers, only one of which is correct. For each correct answer, a candidate gets 1 mark, and no penalty is applied for getting an incorrect answer. A particular candidate answers each question purely by guess-work. Using Normal approximation to Binomial distribution with continuity correction, what is the estimated probability this student obtains a score greater than or equal to 10? Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps. [ANS] A. 0.1615 B. 0.3773 C. 0.7234 D. 0.2766 E. 0.5927", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "An exam consists of 42 multiple-choice questions. Each question has a choice of five answers, only one of which is correct. For each correct answer, a candidate gets 1 mark, and no penalty is applied for getting an incorrect answer. A particular candidate answers each question purely by guess-work. Using Normal approximation to Binomial distribution with continuity correction, what is the estimated probability this student obtains a score greater than or equal to 10? Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps. [ANS] A. 0.5650 B. 0.2089 C. 0.6643 D. 0.4059 E. 0.3357", "answer_v3": [ "E" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0301", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "The lengths of a certain type of chain are approximately Normally distributed with a mean of 3.4 cm and a standard deviation of 0.3 cm.\nFind the value of $\\ell$ such that $P(L > \\ell)=0.01$ [ANS] A. 8.22 cm B. 0.30 cm C. 4.10 cm D. 3.40 cm E. 3.53 cm", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The lengths of a certain type of chain are approximately Normally distributed with a mean of 2.2 cm and a standard deviation of 0.4 cm.\nFind the value of $\\ell$ such that $P(L > \\ell)=0.01$ [ANS] A. 0.40 cm B. 2.37 cm C. 2.20 cm D. 3.13 cm E. 5.53 cm", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "The lengths of a certain type of chain are approximately Normally distributed with a mean of 2.6 cm and a standard deviation of 0.3 cm.\nFind the value of $\\ell$ such that $P(L > \\ell)=0.01$ [ANS] A. 0.30 cm B. 3.30 cm C. 6.36 cm D. 2.60 cm E. 2.73 cm", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0302", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "The time it takes Alice to walk to the bus stop from her home is Normally distributed with mean 13 minutes and variance 3 minutes $^2$. The waiting time for the bus to arrive is Normally distributed with mean 5 minutes and standard deviation 2 minutes. Her bus journey to the UBC bus loop is a Normal variable with mean 24 and standard deviation 5 minutes. The time it take Alice to walk from the bus loop to the lecture theatre to attend STAT 251 is Normally distributed with mean 18 minutes and variance 4 minutes $^2$. The total time taken for Alice to travel from her home to her STAT 251 lecture is Normally distributed. Please use R to find probabilities.\nPart a) What is the mean travel time (in minutes)? [ANS]\nPart b) What is the standard deviation of Alice's travel time (in minutes, to 2 decimal places)? [ANS]\nPart c) The STAT 251 class starts at 8 am sharp. Alice leaves home at 7 am. What is the probability (to 2 decimal places) that Alice will not be late for her class? [ANS]", "answer_v1": [ "60", "6.00", "0.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The time it takes Alice to walk to the bus stop from her home is Normally distributed with mean 11 minutes and variance 2 minutes $^2$. The waiting time for the bus to arrive is Normally distributed with mean 6 minutes and standard deviation 1 minutes. Her bus journey to the UBC bus loop is a Normal variable with mean 24 and standard deviation 5 minutes. The time it take Alice to walk from the bus loop to the lecture theatre to attend STAT 251 is Normally distributed with mean 18 minutes and variance 4 minutes $^2$. The total time taken for Alice to travel from her home to her STAT 251 lecture is Normally distributed. Please use R to find probabilities.\nPart a) What is the mean travel time (in minutes)? [ANS]\nPart b) What is the standard deviation of Alice's travel time (in minutes, to 2 decimal places)? [ANS]\nPart c) The STAT 251 class starts at 8 am sharp. Alice leaves home at 7 am. What is the probability (to 2 decimal places) that Alice will not be late for her class? [ANS]", "answer_v2": [ "59", "5.66", "0.57" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The time it takes Alice to walk to the bus stop from her home is Normally distributed with mean 11 minutes and variance 2 minutes $^2$. The waiting time for the bus to arrive is Normally distributed with mean 5 minutes and standard deviation 2 minutes. Her bus journey to the UBC bus loop is a Normal variable with mean 24 and standard deviation 5 minutes. The time it take Alice to walk from the bus loop to the lecture theatre to attend STAT 251 is Normally distributed with mean 18 minutes and variance 4 minutes $^2$. The total time taken for Alice to travel from her home to her STAT 251 lecture is Normally distributed. Please use R to find probabilities.\nPart a) What is the mean travel time (in minutes)? [ANS]\nPart b) What is the standard deviation of Alice's travel time (in minutes, to 2 decimal places)? [ANS]\nPart c) The STAT 251 class starts at 8 am sharp. Alice leaves home at 7 am. What is the probability (to 2 decimal places) that Alice will not be late for her class? [ANS]", "answer_v3": [ "58", "5.92", "0.63" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0303", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "You purchase a chainsaw, and can buy one of two types of batteries to power it, namely Duxcell and Infinitycell. Batteries of each type have lifetimes before recharge that can be assumed independent and Normally distributed. The mean and standard deviation of the lifetimes of the Duxcell batteries are 10 and 2 minutes respectively, the mean and standard deviation for the Infinitycell batteries are 18 and 3 minutes respectively.\nPart a) What is the probability that a Duxcell battery will last longer than an Infinitycell battery? Give your answer to two decimal places. [ANS]\nPart b) What is the probability that an Infinitycell battery will last more than twice as long as a Duxcell battery? Give your answer to two decimal places. [ANS]\nPart c) You are going to cut down a large tree and do not want to break off from the job to recharge your chainsaw battery. You buy two Duxcell batteries, and plan to use one until it runs out of power, after which you immediately replace it with the second battery. How long (in minutes) can the job last so that with probability 0.75 you can complete the job using the two Duxcell batteries in sequence? Provide your answer to 1 decimal place. [ANS]", "answer_v1": [ "0.0133", "0.3446", "18.1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "You purchase a chainsaw, and can buy one of two types of batteries to power it, namely Duxcell and Infinitycell. Batteries of each type have lifetimes before recharge that can be assumed independent and Normally distributed. The mean and standard deviation of the lifetimes of the Duxcell batteries are 10 and 2 minutes respectively, the mean and standard deviation for the Infinitycell batteries are 11 and 3 minutes respectively.\nPart a) What is the probability that a Duxcell battery will last longer than an Infinitycell battery? Give your answer to two decimal places. [ANS]\nPart b) What is the probability that an Infinitycell battery will last more than twice as long as a Duxcell battery? Give your answer to two decimal places. [ANS]\nPart c) You are going to cut down a large tree and do not want to break off from the job to recharge your chainsaw battery. You buy two Duxcell batteries, and plan to use one until it runs out of power, after which you immediately replace it with the second battery. How long (in minutes) can the job last so that with probability 0.75 you can complete the job using the two Duxcell batteries in sequence? Provide your answer to 1 decimal place. [ANS]", "answer_v2": [ "0.3908", "0.0359", "18.1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "You purchase a chainsaw, and can buy one of two types of batteries to power it, namely Duxcell and Infinitycell. Batteries of each type have lifetimes before recharge that can be assumed independent and Normally distributed. The mean and standard deviation of the lifetimes of the Duxcell batteries are 10 and 2 minutes respectively, the mean and standard deviation for the Infinitycell batteries are 14 and 3 minutes respectively.\nPart a) What is the probability that a Duxcell battery will last longer than an Infinitycell battery? Give your answer to two decimal places. [ANS]\nPart b) What is the probability that an Infinitycell battery will last more than twice as long as a Duxcell battery? Give your answer to two decimal places. [ANS]\nPart c) You are going to cut down a large tree and do not want to break off from the job to recharge your chainsaw battery. You buy two Duxcell batteries, and plan to use one until it runs out of power, after which you immediately replace it with the second battery. How long (in minutes) can the job last so that with probability 0.75 you can complete the job using the two Duxcell batteries in sequence? Provide your answer to 1 decimal place. [ANS]", "answer_v3": [ "0.1336", "0.1151", "18.1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0304", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "The lifetime of lightbulbs that are advertised to last for 5800 hours are normally distributed with a mean of 6000 hours and a standard deviation of 250 hours. What is the probability that a bulb lasts longer than the advertised figure? Probability=[ANS]", "answer_v1": [ "0.788144600433459" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The lifetime of lightbulbs that are advertised to last for 3700 hours are normally distributed with a mean of 4000 hours and a standard deviation of 100 hours. What is the probability that a bulb lasts longer than the advertised figure? Probability=[ANS]", "answer_v2": [ "0.998650101019264" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The lifetime of lightbulbs that are advertised to last for 4400 hours are normally distributed with a mean of 4650 hours and a standard deviation of 150 hours. What is the probability that a bulb lasts longer than the advertised figure? Probability=[ANS]", "answer_v3": [ "0.952209646729261" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0305", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "The top-selling Red and Voss tire is rated 70000 miles, which means nothing. In fact, the distance the tires can run until wear-out is a normally distributed random variable with a mean of 83000 miles and a standard deviation of 6000 miles. A. What is the probability that the tire wears out before 70000 miles? Probability=[ANS]\nB. What is the probability that a tire lasts more than 91000 miles? Probability=[ANS]", "answer_v1": [ "0.0151301390236482", "0.0912112187392869" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The top-selling Red and Voss tire is rated 70000 miles, which means nothing. In fact, the distance the tires can run until wear-out is a normally distributed random variable with a mean of 80000 miles and a standard deviation of 6000 miles. A. What is the probability that the tire wears out before 70000 miles? Probability=[ANS]\nB. What is the probability that a tire lasts more than 86000 miles? Probability=[ANS]", "answer_v2": [ "0.0477903512862291", "0.158655252944872" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The top-selling Red and Voss tire is rated 60000 miles, which means nothing. In fact, the distance the tires can run until wear-out is a normally distributed random variable with a mean of 72000 miles and a standard deviation of 7000 miles. A. What is the probability that the tire wears out before 60000 miles? Probability=[ANS]\nB. What is the probability that a tire lasts more than 82000 miles? Probability=[ANS]", "answer_v3": [ "0.0432381317602467", "0.0765637245232524" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0306", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "College students average 8.5 hours of sleep per night with a standard deviation of 40 minutes. If the amount of sleep is normally distributed, what proportion of college students sleep for more than 9.6 hours? Proportion=[ANS]", "answer_v1": [ "0.0494714670470627" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "College students average 7.1 hours of sleep per night with a standard deviation of 45 minutes. If the amount of sleep is normally distributed, what proportion of college students sleep for more than 7.9 hours? Proportion=[ANS]", "answer_v2": [ "0.143061191208927" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "College students average 7.6 hours of sleep per night with a standard deviation of 40 minutes. If the amount of sleep is normally distributed, what proportion of college students sleep for more than 8.5 hours? Proportion=[ANS]", "answer_v3": [ "0.0885079904508209" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0307", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "Because of the relatively high interest rates, most consumers attempt to pay off their credit card bills promptly. However, this is not always possible. An analysis of the amount of interest paid monthly by a bank's Visa cardholders reveals that the amount is normally distributed with a mean of 29 dollars and a standard deviation of 8 dollars. A. What proportion of the bank's Visa cardholders pay more than 32 dollars in interest? Proportion=[ANS]\nB. What proportion of the bank's Visa cardholders pay more than 40 dollars in interest? Proportion=[ANS]\nC. What proportion of the bank's Visa cardholders pay less than 19 dollars in interest? Proportion=[ANS]\nD. What interest payment is exceeded by only 19\\% of the bank's Visa cardholders? Interest Payment=[ANS]", "answer_v1": [ "0.353830232340497", "0.0845657213647544", "0.105649772680275", "36.023168" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Because of the relatively high interest rates, most consumers attempt to pay off their credit card bills promptly. However, this is not always possible. An analysis of the amount of interest paid monthly by a bank's Visa cardholders reveals that the amount is normally distributed with a mean of 25 dollars and a standard deviation of 9 dollars. A. What proportion of the bank's Visa cardholders pay more than 27 dollars in interest? Proportion=[ANS]\nB. What proportion of the bank's Visa cardholders pay more than 35 dollars in interest? Proportion=[ANS]\nC. What proportion of the bank's Visa cardholders pay less than 13 dollars in interest? Proportion=[ANS]\nD. What interest payment is exceeded by only 19\\% of the bank's Visa cardholders? Interest Payment=[ANS]", "answer_v2": [ "0.412070446884092", "0.133260261915925", "0.0912112187392871", "32.901064" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Because of the relatively high interest rates, most consumers attempt to pay off their credit card bills promptly. However, this is not always possible. An analysis of the amount of interest paid monthly by a bank's Visa cardholders reveals that the amount is normally distributed with a mean of 26 dollars and a standard deviation of 8 dollars. A. What proportion of the bank's Visa cardholders pay more than 28 dollars in interest? Proportion=[ANS]\nB. What proportion of the bank's Visa cardholders pay more than 37 dollars in interest? Proportion=[ANS]\nC. What proportion of the bank's Visa cardholders pay less than 17 dollars in interest? Proportion=[ANS]\nD. What interest payment is exceeded by only 19\\% of the bank's Visa cardholders? Interest Payment=[ANS]", "answer_v3": [ "0.401293673330236", "0.0845657213647544", "0.130294516150229", "33.023168" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0308", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "probability", "continuous", "normal", "distribution" ], "problem_v1": "A new car that is a gas-and electric-powered hybrid has recently hit the market. The distance travelled on 1 gallon of fuel is normally distributed with a mean of 60 miles and a standard deviation of 7 miles. Find the probability of the following events: A. The car travels more than 66 miles per gallon. Probability=[ANS]\nB. The car travels less than 53 miles per gallon. Probability=[ANS]\nC. The car travels between 56 and 64 miles per gallon. Probability=[ANS]", "answer_v1": [ "0.195682968167176", "0.158655252944872", "0.432290833802647" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A new car that is a gas-and electric-powered hybrid has recently hit the market. The distance travelled on 1 gallon of fuel is normally distributed with a mean of 45 miles and a standard deviation of 8 miles. Find the probability of the following events: A. The car travels more than 48 miles per gallon. Probability=[ANS]\nB. The car travels less than 40 miles per gallon. Probability=[ANS]\nC. The car travels between 41 and 53 miles per gallon. Probability=[ANS]", "answer_v2": [ "0.353830232340497", "0.26598552806204", "0.532807207342556" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A new car that is a gas-and electric-powered hybrid has recently hit the market. The distance travelled on 1 gallon of fuel is normally distributed with a mean of 50 miles and a standard deviation of 7 miles. Find the probability of the following events: A. The car travels more than 54 miles per gallon. Probability=[ANS]\nB. The car travels less than 44 miles per gallon. Probability=[ANS]\nC. The car travels between 45 and 54 miles per gallon. Probability=[ANS]", "answer_v3": [ "0.283854582111994", "0.195682968167176", "0.478620154874347" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0309", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "The number of pizzas consumed per month by university students is normally distributed with a mean of 12 and a standard deviation of 4. A. What proportion of students consume more than 15 pizzas per month? Probability=[ANS]\nB. What is the probability that in a random sample of size 11, a total of more than 121 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 11 students?) Probability=[ANS]", "answer_v1": [ "0.226627351390247", "0.796491985381333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The number of pizzas consumed per month by university students is normally distributed with a mean of 6 and a standard deviation of 5. A. What proportion of students consume more than 8 pizzas per month? Probability=[ANS]\nB. What is the probability that in a random sample of size 8, a total of more than 64 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 8 students?) Probability=[ANS]", "answer_v2": [ "0.34457825740291", "0.12894951665959" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The number of pizzas consumed per month by university students is normally distributed with a mean of 8 and a standard deviation of 4. A. What proportion of students consume more than 10 pizzas per month? Probability=[ANS]\nB. What is the probability that in a random sample of size 9, a total of more than 63 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 9 students?) Probability=[ANS]", "answer_v3": [ "0.308537537739272", "0.773372646639672" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0310", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "2", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "A professor of statistics noticed that the marks in his course are normally distributed. He also noticed that his morning classes average 73\\% with a standard deviation of 12\\% on their final exams. His afternoon classes average 79\\% with a standard deviation of 9\\%. A. What is the probability that a randomly selected student in the morning class has a higher final exam mark than a randomly selected student from an afternoon class? Probability=[ANS]\nB. What is the probability that the mean mark of four randomly selected students from a morning class is greater than the average mark of four randomly selected students from an afternoon class? Probability=[ANS]", "answer_v1": [ "0.34457825740291", "0.211855397596787" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A professor of statistics noticed that the marks in his course are normally distributed. He also noticed that his morning classes average 75\\% with a standard deviation of 9\\% on their final exams. His afternoon classes average 77\\% with a standard deviation of 12\\%. A. What is the probability that a randomly selected student in the morning class has a higher final exam mark than a randomly selected student from an afternoon class? Probability=[ANS]\nB. What is the probability that the mean mark of four randomly selected students from a morning class is greater than the average mark of four randomly selected students from an afternoon class? Probability=[ANS]", "answer_v2": [ "0.446964882389515", "0.394862909477193" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A professor of statistics noticed that the marks in his course are normally distributed. He also noticed that his morning classes average 73\\% with a standard deviation of 10\\% on their final exams. His afternoon classes average 78\\% with a standard deviation of 9\\%. A. What is the probability that a randomly selected student in the morning class has a higher final exam mark than a randomly selected student from an afternoon class? Probability=[ANS]\nB. What is the probability that the mean mark of four randomly selected students from a morning class is greater than the average mark of four randomly selected students from an afternoon class? Probability=[ANS]", "answer_v3": [ "0.355077817101494", "0.22865180643604" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0311", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "The time it takes for a statistics professor to mark a single midterm test is normally distributed with a mean of 4.9 minutes and a standard deviation of 2.1 minutes. There are 61 students in the professor's class. What is the probability that he needs more than 5 hours to mark all of the midterm tests? Probability=[ANS]", "answer_v1": [ "0.473264206501443" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The time it takes for a statistics professor to mark a single midterm test is normally distributed with a mean of 4.6 minutes and a standard deviation of 2.5 minutes. There are 56 students in the professor's class. What is the probability that he needs more than 5 hours to mark all of the midterm tests? Probability=[ANS]", "answer_v2": [ "0.0117142053405121" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The time it takes for a statistics professor to mark a single midterm test is normally distributed with a mean of 4.7 minutes and a standard deviation of 2.3 minutes. There are 58 students in the professor's class. What is the probability that he needs more than 5 hours to mark all of the midterm tests? Probability=[ANS]", "answer_v3": [ "0.0588783486564563" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0312", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "4", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "A factory's worker productivity is normally distributed. One worker produces an average of 75 units per day with a standard deviation of 21. Another worker produces at an average rate of 67 units per day with a standard deviation of 20. A. What is the probability that in a single day worker 1 will outproduce worker 2? Probability=[ANS]\nB. What is the probability that during one week (5 working days), worker 1 will outproduce worker 2? Probability=[ANS]", "answer_v1": [ "0.608672994806331", "0.731331956947477" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A factory's worker productivity is normally distributed. One worker produces an average of 77 units per day with a standard deviation of 18. Another worker produces at an average rate of 65 units per day with a standard deviation of 24. A. What is the probability that in a single day worker 1 will outproduce worker 2? Probability=[ANS]\nB. What is the probability that during one week (5 working days), worker 1 will outproduce worker 2? Probability=[ANS]", "answer_v2": [ "0.655421740624574", "0.814453314255974" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A factory's worker productivity is normally distributed. One worker produces an average of 75 units per day with a standard deviation of 19. Another worker produces at an average rate of 66 units per day with a standard deviation of 20. A. What is the probability that in a single day worker 1 will outproduce worker 2? Probability=[ANS]\nB. What is the probability that during one week (5 working days), worker 1 will outproduce worker 2? Probability=[ANS]", "answer_v3": [ "0.627882295350458", "0.767157142722448" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0313", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "Suppose that the number of customers who enter a supermarket each hour is normally distributed with a mean of 650 and a standard deviation of 210. The supermarket is open 16 hours per day. What is the probability that the total number of customers who enter the supermarket in one day is greater than 11000? (Hint: Calculate the average hourly number of customers necessary to exceed 11000 in one 16-hour day.) Probability=[ANS]", "answer_v1": [ "0.237525261040346" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the number of customers who enter a supermarket each hour is normally distributed with a mean of 510 and a standard deviation of 250. The supermarket is open 13 hours per day. What is the probability that the total number of customers who enter the supermarket in one day is greater than 6000? (Hint: Calculate the average hourly number of customers necessary to exceed 6000 in one 13-hour day.) Probability=[ANS]", "answer_v2": [ "0.757699687344447" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the number of customers who enter a supermarket each hour is normally distributed with a mean of 560 and a standard deviation of 210. The supermarket is open 13 hours per day. What is the probability that the total number of customers who enter the supermarket in one day is greater than 7400? (Hint: Calculate the average hourly number of customers necessary to exceed 7400 in one 13-hour day.) Probability=[ANS]", "answer_v3": [ "0.437037015298626" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0314", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "The manager of a restaurant believes that waiters and waitresses who introduce themselves by telling customers their names will get larger tips than those who don't. In fact, she claims that the average tip for the former group is 17\\% while that of the latter is only 13\\%. If tips are normally distributed with a standard deviation of 9\\%, what is the probability that in a random sample of 14 tips recorded from waiters and waitresses who introduce themselves and 14 tips from waiters and waitresses who don't, the mean of the former will exceed that of the latter? Probability=[ANS]", "answer_v1": [ "0.880180475712614" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The manager of a restaurant believes that waiters and waitresses who introduce themselves by telling customers their names will get larger tips than those who don't. In fact, she claims that the average tip for the former group is 18\\% while that of the latter is only 12\\%. If tips are normally distributed with a standard deviation of 8\\%, what is the probability that in a random sample of 10 tips recorded from waiters and waitresses who introduce themselves and 10 tips from waiters and waitresses who don't, the mean of the former will exceed that of the latter? Probability=[ANS]", "answer_v2": [ "0.953233742657054" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The manager of a restaurant believes that waiters and waitresses who introduce themselves by telling customers their names will get larger tips than those who don't. In fact, she claims that the average tip for the former group is 17\\% while that of the latter is only 12\\%. If tips are normally distributed with a standard deviation of 8\\%, what is the probability that in a random sample of 11 tips recorded from waiters and waitresses who introduce themselves and 11 tips from waiters and waitresses who don't, the mean of the former will exceed that of the latter? Probability=[ANS]", "answer_v3": [ "0.928642466543703" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0315", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Application of a normal distribution", "level": "3", "keywords": [ "statistics", "sampling distributions", "normal sampling distributions" ], "problem_v1": "An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean of 118 cm and a standard deviation of 5.3 cm. A. Find the probability that one selected subcomponent is longer than 120 cm. Probability=[ANS]\nB. Find the probability that if 4 subcomponents are randomly selected, their mean length exceeds 120 cm. Probability=[ANS]\nC. Find the probability that if 4 are randomly selected, all 4 have lengths that exceed 120 cm. Probability=[ANS]", "answer_v1": [ "0.352953604139577", "0.225209406703194", "0.0155192412356597" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean of 110 cm and a standard deviation of 5.7 cm. A. Find the probability that one selected subcomponent is longer than 112 cm. Probability=[ANS]\nB. Find the probability that if 3 subcomponents are randomly selected, their mean length exceeds 112 cm. Probability=[ANS]\nC. Find the probability that if 3 are randomly selected, all 3 have lengths that exceed 112 cm. Probability=[ANS]", "answer_v2": [ "0.362840240204242", "0.271680916945863", "0.0477690206250974" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean of 113 cm and a standard deviation of 5.3 cm. A. Find the probability that one selected subcomponent is longer than 115 cm. Probability=[ANS]\nB. Find the probability that if 3 subcomponents are randomly selected, their mean length exceeds 115 cm. Probability=[ANS]\nC. Find the probability that if 3 are randomly selected, all 3 have lengths that exceed 115 cm. Probability=[ANS]", "answer_v3": [ "0.352953604139577", "0.256683457032257", "0.0439696352541637" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0316", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "3", "keywords": [ "Probability Density Function", "PDF", "Continuous", "Expectation", "Expected Value" ], "problem_v1": "The density function of $X$ is given by f(x)=\\begin{cases} a+b x^2 & \\mbox{if} 0\\le x\\le 1\\\\ 0 & \\mbox{otherwise}\\end{cases}. If the expectation of $X$ is $E(X)=0.4375$, find $a$ and $b$. $a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "1.25", "-0.75" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The density function of $X$ is given by f(x)=\\begin{cases} a+b x^2 & \\mbox{if} 0\\le x\\le 1\\\\ 0 & \\mbox{otherwise}\\end{cases}. If the expectation of $X$ is $E(X)=0.75$, find $a$ and $b$. $a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "0", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The density function of $X$ is given by f(x)=\\begin{cases} a+b x^2 & \\mbox{if} 0\\le x\\le 1\\\\ 0 & \\mbox{otherwise}\\end{cases}. If the expectation of $X$ is $E(X)=0.625$, find $a$ and $b$. $a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "0.5", "1.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0317", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "3", "keywords": [ "statistics", "inference", "hypothesis testing", "t score" ], "problem_v1": "What is the value of t*, the critical value of the t distribution for a sample of size 22, such that the probability of being greater than t*or less than-t*is 1\\%? t*=[ANS]", "answer_v1": [ "2.831" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the value of t*, the critical value of the t distribution for a sample of size 6, such that the probability of being greater than t*or less than-t*is 1\\%? t*=[ANS]", "answer_v2": [ "4.032" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the value of t*, the critical value of the t distribution for a sample of size 10, such that the probability of being greater than t*or less than-t*is 1\\%? t*=[ANS]", "answer_v3": [ "3.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0318", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "3", "keywords": [ "percent" ], "problem_v1": "For an F-curve with degrees of freedom df=(12,5) find the F-value that has area 0.01 to its right. [ANS] A. 5.06 B. 3.27 C. 9.89 D. None of the above\nFor an F-curve with degrees of freedom df=(4,10) find the F-value that has area 0.005 to its right [ANS] A. 7.34 B. 20.97 C. 2.61 D. None of the above", "answer_v1": [ "C", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "For an F-curve with degrees of freedom df=(12,5) find the F-value that has area 0.01 to its right. [ANS] A. 9.89 B. 3.27 C. 5.06 D. None of the above\nFor an F-curve with degrees of freedom df=(4,10) find the F-value that has area 0.005 to its right [ANS] A. 7.34 B. 2.61 C. 20.97 D. None of the above", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "For an F-curve with degrees of freedom df=(12,5) find the F-value that has area 0.01 to its right. [ANS] A. 5.06 B. 9.89 C. 3.27 D. None of the above\nFor an F-curve with degrees of freedom df=(4,10) find the F-value that has area 0.005 to its right [ANS] A. 20.97 B. 7.34 C. 2.61 D. None of the above", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0319", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "4", "keywords": [ "Random variables", "function of a continuous random variable", "expectation", "variance", "mean and variance of a non-linear function of a continuous random variable" ], "problem_v1": "The time, in 100 hours, that a student uses her game console over a year is a random variable $X$ with probability density function $f(x)=\\begin{cases} x \\text{if} 0 < x < 1 \\\\ 2-x \\text{if} 1 \\leq x < 2 \\\\ 0 \\text{otherwise.} \\end{cases}$ The power (in number of kilowatt hours) expended by the student's game console each year is $44X^2+22$. For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) Find the mean amount of power expended by the student's game console per year. [ANS]\nPart b) Find the variance of power expended by the student's game console per year. [ANS]", "answer_v1": [ "73.333", "1365.96" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The time, in 100 hours, that a student uses her game console over a year is a random variable $X$ with probability density function $f(x)=\\begin{cases} x \\text{if} 0 < x < 1 \\\\ 2-x \\text{if} 1 \\leq x < 2 \\\\ 0 \\text{otherwise.} \\end{cases}$ The power (in number of kilowatt hours) expended by the student's game console each year is $50 X^2+28$. For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) Find the mean amount of power expended by the student's game console per year. [ANS]\nPart b) Find the variance of power expended by the student's game console per year. [ANS]", "answer_v2": [ "86.333", "1763.89" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The time, in 100 hours, that a student uses her game console over a year is a random variable $X$ with probability density function $f(x)=\\begin{cases} x \\text{if} 0 < x < 1 \\\\ 2-x \\text{if} 1 \\leq x < 2 \\\\ 0 \\text{otherwise.} \\end{cases}$ The power (in number of kilowatt hours) expended by the student's game console each year is $48X^2+26$. For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) Find the mean amount of power expended by the student's game console per year. [ANS]\nPart b) Find the variance of power expended by the student's game console per year. [ANS]", "answer_v3": [ "82", "1625.6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0320", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "3", "keywords": [ "Random variables", "cumulative distribution functions", "expectation", "determination of cumulative distribution function", "finding the mean given the cumulative distribution function", "finding the proportion of distribution less than the mean" ], "problem_v1": "The annual salaries (in \\$) within a certain profession are modelled by a random variable with the cumulative distribution function\n$F(x)=\\begin{cases} 1-k x^{-3} \\text{for} x > 46000 \\\\ 0 \\text{otherwise,} \\end{cases}$ for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) Find the constant k here and provide its natural logarithm to three decimal places.\nNatural logarithm of k: [ANS]\nPart b) Calculate the mean salary given by the model. [ANS]\nPart c) Find the proportion in the profession earning less than the mean, giving your answers as a fraction or to three decimal places. [ANS]", "answer_v1": [ "32.209", "69000", "0.704" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The annual salaries (in \\$) within a certain profession are modelled by a random variable with the cumulative distribution function\n$F(x)=\\begin{cases} 1-k x^{-3} \\text{for} x > 40000 \\\\ 0 \\text{otherwise,} \\end{cases}$ for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) Find the constant k here and provide its natural logarithm to three decimal places.\nNatural logarithm of k: [ANS]\nPart b) Calculate the mean salary given by the model. [ANS]\nPart c) Find the proportion in the profession earning less than the mean, giving your answers as a fraction or to three decimal places. [ANS]", "answer_v2": [ "31.79", "60000", "0.704" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The annual salaries (in \\$) within a certain profession are modelled by a random variable with the cumulative distribution function\n$F(x)=\\begin{cases} 1-k x^{-3} \\text{for} x > 52000 \\\\ 0 \\text{otherwise,} \\end{cases}$ for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) Find the constant k here and provide its natural logarithm to three decimal places.\nNatural logarithm of k: [ANS]\nPart b) Calculate the mean salary given by the model. [ANS]\nPart c) Find the proportion in the profession earning less than the mean, giving your answers as a fraction or to three decimal places. [ANS]", "answer_v3": [ "32.577", "78000", "0.704" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0321", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "3", "keywords": [ "Random variables", "excess lifetime/reliability", "expectation", "finding a cumulative probability", "inversion of c.d.f. to find value exceeded with a given probability", "determination of density function and mean given excess lifetime distribution/reliability" ], "problem_v1": "The time that a butterfly lives after emerging from its chrysalis can be modelled by a random variable $T$, the model here taking the probability that a butterfly survives for more than $t$ days as\n$P(T > t)=\\frac{36}{(6+t)^2}, \\text{} t \\geq 0.$ For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) What is the probability that a butterfly will die within 7 days of emerging? [ANS]\nPart b) If a large number of butterflies emerge on the same day, after how many days would you expect only 7 \\% to be alive? [ANS]\nPart c) Calculate the mean lifetime of a butterfly after emerging from its chrysalis. [ANS]", "answer_v1": [ "0.787", "16.678", "6" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The time that a butterfly lives after emerging from its chrysalis can be modelled by a random variable $T$, the model here taking the probability that a butterfly survives for more than $t$ days as\n$P(T > t)=\\frac{36}{(6+t)^2}, \\text{} t \\geq 0.$ For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) What is the probability that a butterfly will die within 4 days of emerging? [ANS]\nPart b) If a large number of butterflies emerge on the same day, after how many days would you expect only 4 \\% to be alive? [ANS]\nPart c) Calculate the mean lifetime of a butterfly after emerging from its chrysalis. [ANS]", "answer_v2": [ "0.64", "24", "6" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The time that a butterfly lives after emerging from its chrysalis can be modelled by a random variable $T$, the model here taking the probability that a butterfly survives for more than $t$ days as\n$P(T > t)=\\frac{36}{(6+t)^2}, \\text{} t \\geq 0.$ For these problems, please ensure your answers are accurate to within 3 decimals.\nPart a) What is the probability that a butterfly will die within 5 days of emerging? [ANS]\nPart b) If a large number of butterflies emerge on the same day, after how many days would you expect only 5 \\% to be alive? [ANS]\nPart c) Calculate the mean lifetime of a butterfly after emerging from its chrysalis. [ANS]", "answer_v3": [ "0.702", "20.833", "6" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0322", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "4", "keywords": [ "statistics", "continuous random variables", "probability", "uniform distribution" ], "problem_v1": "You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned to write a portion of computer code. Both of you work simultaneously, but independently. The completion time of your task follows a uniform distribution between 30 and 50 hours. Your partner is stronger in programming and his task is more complex, and the completion time for his task follows a uniform distribution between 37 and 54 hours.\nPart a) What is the expected completion time (in hours) for your partner's task? [ANS] A. 40 B. 51.5 C. 45.5 D. 48 E. 53\nPart b) What is the corresponding standard deviation for the completion time of your partner's task? [ANS] A. 24.08 B. 4.91 C. 2.31 D. 6.00 E. 1.19\nPart c) What is the probability that you and your partner are not able to hand in your project on time (that is, your team's project completion time exceeds 48 hours)? [ANS] A. 0.4176 B. 0.4888 C. 0.5455 D. 0.6823 E. 0.5200\nPart d) On the 48th hour when the project is due, you and your partner have not completed the project. You approach the course instructor for an extension. The course instructor grants you and your partner an extension of 4 hours to hand in the project starting from the 48th hour. What is the probability you and your partner are now able to meet the new deadline? [ANS] A. 0.7183 B. 0.4375 C. 0.6888 D. 0.775 E. 0.3175", "answer_v1": [ "C", "B", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned to write a portion of computer code. Both of you work simultaneously, but independently. The completion time of your task follows a uniform distribution between 30 and 50 hours. Your partner is stronger in programming and his task is more complex, and the completion time for his task follows a uniform distribution between 31 and 56 hours.\nPart a) What is the expected completion time (in hours) for your partner's task? [ANS] A. 51.5 B. 53 C. 48 D. 43.5 E. 40\nPart b) What is the corresponding standard deviation for the completion time of your partner's task? [ANS] A. 3.34 B. 52.08 C. 7.22 D. 1.44 E. 19.58\nPart c) What is the probability that you and your partner are not able to hand in your project on time (that is, your team's project completion time exceeds 48 hours)? [ANS] A. 0.4888 B. 0.5455 C. 0.3880 D. 0.5200 E. 0.6823\nPart d) On the 48th hour when the project is due, you and your partner have not completed the project. You approach the course instructor for an extension. The course instructor grants you and your partner an extension of 4 hours to hand in the project starting from the 48th hour. What is the probability you and your partner are now able to meet the new deadline? [ANS] A. 0.5876 B. 0.3175 C. 0.775 D. 0.4375 E. 0.6888", "answer_v2": [ "D", "C", "C", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned to write a portion of computer code. Both of you work simultaneously, but independently. The completion time of your task follows a uniform distribution between 30 and 50 hours. Your partner is stronger in programming and his task is more complex, and the completion time for his task follows a uniform distribution between 33 and 55 hours.\nPart a) What is the expected completion time (in hours) for your partner's task? [ANS] A. 51.5 B. 44 C. 40 D. 48 E. 53\nPart b) What is the corresponding standard deviation for the completion time of your partner's task? [ANS] A. 40.33 B. 2.97 C. 1.35 D. 6.35 E. 17.20\nPart c) What is the probability that you and your partner are not able to hand in your project on time (that is, your team's project completion time exceeds 48 hours)? [ANS] A. 0.6823 B. 0.4888 C. 0.5455 D. 0.3864 E. 0.5200\nPart d) On the 48th hour when the project is due, you and your partner have not completed the project. You approach the course instructor for an extension. The course instructor grants you and your partner an extension of 4 hours to hand in the project starting from the 48th hour. What is the probability you and your partner are now able to meet the new deadline? [ANS] A. 0.6471 B. 0.6888 C. 0.3175 D. 0.775 E. 0.4375", "answer_v3": [ "B", "D", "D", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0323", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "3", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "A random variable $X$ follows a Normal distribution with mean $\\mu=43$ and standard deviation $\\sigma=4$.\nWhich of the following gives the expectation $E(X^2)$? [ANS] A. 2209 B. 1833 C. 1865 D. 1849 E. Insufficient information to calculate $E(X^2)$", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "A random variable $X$ follows a Normal distribution with mean $\\mu=35$ and standard deviation $\\sigma=6$.\nWhich of the following gives the expectation $E(X^2)$? [ANS] A. 1681 B. 1189 C. 1225 D. 1261 E. Insufficient information to calculate $E(X^2)$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "A random variable $X$ follows a Normal distribution with mean $\\mu=38$ and standard deviation $\\sigma=4$.\nWhich of the following gives the expectation $E(X^2)$? [ANS] A. 1764 B. 1460 C. 1444 D. 1428 E. Insufficient information to calculate $E(X^2)$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0324", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "3", "keywords": [ "statistics", "Inference", "inference about a population" ], "problem_v1": "Which of the following statements is false? [ANS] A. As the degrees of freedom get smaller, the t-distribution's dispersion gets smaller. B. The t-distribution is mound-shaped C. The t-distribution is more spread out than the standard normal distribution. D. The t-distribution is symmetric about zero.\nThe Student t-distribution approaches the normal distribution as the: [ANS] A. population size increases B. degrees of freedom decrease C. degrees of freedom increase D. sample size decreases", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Which of the following statements is false? [ANS] A. The t-distribution is more spread out than the standard normal distribution. B. The t-distribution is mound-shaped C. The t-distribution is symmetric about zero. D. As the degrees of freedom get smaller, the t-distribution's dispersion gets smaller.\nA robust estimator is one that: [ANS] A. is not sensitive to a moderate departure from the assumption of normality in the population B. is efficient and less spread out C. is unbiased and symmetrical about zero D. is consistent and is also mound-shaped", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Which of the following statements is false? [ANS] A. The t-distribution is more spread out than the standard normal distribution. B. As the degrees of freedom get smaller, the t-distribution's dispersion gets smaller. C. The t-distribution is mound-shaped D. The t-distribution is symmetric about zero.\nFor statistical inference about the mean of a single population when the population standard deviation is unknown, the degrees of freedom for the t-distribution equal $n-1$ because we lose one degree of freedom by using the: [ANS] A. sample mean as an estimate of the population mean B. sample proportion as an estimate of the population proportion C. sample standard deviation as an estimate of the population standard deviation D. sample size as an estimate of the population size", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0325", "subject": "Probability", "topic": "Continuous distributions", "subtopic": "Other distribution", "level": "2", "keywords": [ "statistics", "Inference", "inference about a population" ], "problem_v1": "Like that of the Student t-distribution, the shape of the chi-squared distribution depends on: [ANS] A. the number of its degrees of freedom B. whether the population is unimodal or bimodal C. the population standard deviation D. the population size\nThe statistic $(n-1)s^2/\\sigma^2$ is chi-square distributed with $n-1$ degrees of freedom only if [ANS] A. the population is normally distributed with variance equal to $\\sigma^2$ B. the sample has a Student t-distribution with degrees of freedom equal to $n-1$ C. the sample is normally distributed with variance equal to $s^2$ D. all of the above statements are correct", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Like that of the Student t-distribution, the shape of the chi-squared distribution depends on: [ANS] A. the population standard deviation B. whether the population is unimodal or bimodal C. the population size D. the number of its degrees of freedom\nThe chi-squared distribution is: [ANS] A. positively skewed ranging between $0$ and $\\infty$ B. mound-shaped C. symmetrical about 0 D. negatively skewed ranging between $-\\infty$ and $0$", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Like that of the Student t-distribution, the shape of the chi-squared distribution depends on: [ANS] A. the population standard deviation B. the number of its degrees of freedom C. whether the population is unimodal or bimodal D. the population size\nFor a sample of size 20 taken from a normally distributed population with standard deviation equal to 5, a 90\\% confidence interval for the population mean would require the use of: [ANS] A. $z=1.645$ B. $t=1.729$ C. $t=1.328$ D. $\\chi^2=11.6509$", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Probability_0326", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Chebychev's inequality", "level": "2", "keywords": [], "problem_v1": "A statistician uses Chebyshev's Theorem to estimate that at least 75 \\% of a population lies between the values 7 and 16. Use this information to find the values of the population mean, $\\mu$, and the population standard deviation $\\sigma$. a) $\\mu=$ [ANS]\nb) $\\sigma=$ [ANS]", "answer_v1": [ "11.5", "2.25" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A statistician uses Chebyshev's Theorem to estimate that at least 10 \\% of a population lies between the values 2 and 20. Use this information to find the values of the population mean, $\\mu$, and the population standard deviation $\\sigma$. a) $\\mu=$ [ANS]\nb) $\\sigma=$ [ANS]", "answer_v2": [ "11", "8.53814968245462" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A statistician uses Chebyshev's Theorem to estimate that at least 30 \\% of a population lies between the values 3 and 17. Use this information to find the values of the population mean, $\\mu$, and the population standard deviation $\\sigma$. a) $\\mu=$ [ANS]\nb) $\\sigma=$ [ANS]", "answer_v3": [ "10", "5.85662018573853" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0327", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Chebychev's inequality", "level": "2", "keywords": [], "problem_v1": "Suppose that the blood pressure of the human inhabitants of a certain Pacific island is distributed with mean, $\\mu$=107 mmHg and standard deviation, $\\sigma$=11 mmHg. According to Chebyshev's Theorem, at least what percentage of the islander's have blood pressure in the range from 75.1 mmHg to 138.9 mmHg? answer: [ANS] \\%", "answer_v1": [ "88.1093935790725" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the blood pressure of the human inhabitants of a certain Pacific island is distributed with mean, $\\mu$=69 mmHg and standard deviation, $\\sigma$=16 mmHg. According to Chebyshev's Theorem, at least what percentage of the islander's have blood pressure in the range from 40.2 mmHg to 97.8 mmHg? answer: [ANS] \\%", "answer_v2": [ "69.1358024691358" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the blood pressure of the human inhabitants of a certain Pacific island is distributed with mean, $\\mu$=82 mmHg and standard deviation, $\\sigma$=12 mmHg. According to Chebyshev's Theorem, at least what percentage of the islander's have blood pressure in the range from 56.8 mmHg to 107.2 mmHg? answer: [ANS] \\%", "answer_v3": [ "77.3242630385488" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0328", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Chebychev's inequality", "level": "3", "keywords": [ "Random variables", "Chebychev Inequality", "given the mean and standard deviation finding an upper bound on the probability a variable is not within a given distance from its mean", "and a lower bound on the probability the variable falls a given distance from its mean" ], "problem_v1": "An instructor believes the distribution of scores on her final exam will have mean $69$ and standard deviation $8$. Fred is a student in the course that was selected at random. Provide answers to the following to two decimal places.\n(a) Find an upper bound on the probability that Fred will not score between $54$ and $84$. [ANS]\n(b) Provide a lower bound on the probability that Fred will score between $50$ and $88$. [ANS]", "answer_v1": [ "0.28", "0.82" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An instructor believes the distribution of scores on her final exam will have mean $61$ and standard deviation $10$. Fred is a student in the course that was selected at random. Provide answers to the following to two decimal places.\n(a) Find an upper bound on the probability that Fred will not score between $46$ and $76$. [ANS]\n(b) Provide a lower bound on the probability that Fred will score between $45$ and $77$. [ANS]", "answer_v2": [ "0.44", "0.61" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An instructor believes the distribution of scores on her final exam will have mean $64$ and standard deviation $8$. Fred is a student in the course that was selected at random. Provide answers to the following to two decimal places.\n(a) Find an upper bound on the probability that Fred will not score between $49$ and $79$. [ANS]\n(b) Provide a lower bound on the probability that Fred will score between $47$ and $81$. [ANS]", "answer_v3": [ "0.28", "0.78" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0329", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Chebychev's inequality", "level": "2", "keywords": [ "statistics", "descriptive statistics" ], "problem_v1": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. The length of the box in a box-and-whisker plot portrays the interquartile range. [ANS] 2. In a histogram, the proportion of the total area which must be to the left of the median is less than 0.50 if the distribution is skewed to the left. [ANS] 3. While Chebyshev's theorem applies to any distribution, regardless of shape, the Empirical Rule applies only to distributions that are bell shaped and symmetrical. [ANS] 4. The mean of fifty sales receipts is 66.75 and the standard deviation is 10.55. Using Chebyshev's theorem, at least 75\\% of the sales receipts were between 45.65 and 87.85", "answer_v1": [ "T", "F", "T", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. In a histogram, the proportion of the total area which must be to the left of the median is less than 0.50 if the distribution is skewed to the left. [ANS] 2. The mean of fifty sales receipts is 66.75 and the standard deviation is 10.55. Using Chebyshev's theorem, at least 75\\% of the sales receipts were between 45.65 and 87.85 [ANS] 3. The length of the box in a box-and-whisker plot portrays the interquartile range. [ANS] 4. According to Chebyshev's theorem, at least 93.75\\% of observations should fall within 4 standard deviations of the mean.", "answer_v2": [ "F", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Select True or False from each pull-down menu, depending on whether the corresponding statement is true or false.\n[ANS] 1. Chebyshev's theorem states that the percentage of observations in a data set that should fall within five standard deviations of the mean is at least 96\\%. [ANS] 2. According to Chebyshev's theorem, at least 93.75\\% of observations should fall within 4 standard deviations of the mean. [ANS] 3. The length of the box in a box-and-whisker plot portrays the interquartile range. [ANS] 4. Chebyshev's theorem applies only to data sets that have a bell shaped distribution.", "answer_v3": [ "T", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Probability_0330", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Chebychev's inequality", "level": "2", "keywords": [ "statistics", "descriptive statistics" ], "problem_v1": "The Empirical Rule states that the approximate percentage of measurements in a data set (providing that the data set has a bell-shaped distribution) that fall within two standard deviations of their mean is approximately: [ANS] A. 95\\% B. 99\\% C. 75\\% D. 68\\%\nWhich of the following summary measures is affected most by outliers? [ANS] A. The interquartile range B. The median C. The range D. The geometric mean E. All of the above", "answer_v1": [ "A", "C" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "The Empirical Rule states that the approximate percentage of measurements in a data set (providing that the data set has a bell-shaped distribution) that fall within two standard deviations of their mean is approximately: [ANS] A. 75\\% B. 99\\% C. 68\\% D. 95\\%\nChebyshevs Theorem states that the percentage of measurements in a data set that fall within three standard deviations of their mean is: [ANS] A. at least 89\\% B. 89\\% C. 75\\% D. at least 75\\%", "answer_v2": [ "D", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "The Empirical Rule states that the approximate percentage of measurements in a data set (providing that the data set has a bell-shaped distribution) that fall within two standard deviations of their mean is approximately: [ANS] A. 75\\% B. 95\\% C. 99\\% D. 68\\%\nWhich of the following summary measures cannot be easily approximated from a box-and-whisker plot? [ANS] A. The standard deviation B. The interquartile range C. The range D. The second quartile E. All of the above", "answer_v3": [ "B", "A" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Probability_0331", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Central limit theorem", "level": "3", "keywords": [ "Probability", "Central Limit Theorem" ], "problem_v1": "$85$ numbers are rounded off to the nearest integer and then summed. If the individual round-off error are uniformly distributed over $(-.5,.5)$ what is the probability that the resultant sum differs from the exact sum by more than $4$? [ANS]", "answer_v1": [ "0.132854955731054" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$45$ numbers are rounded off to the nearest integer and then summed. If the individual round-off error are uniformly distributed over $(-.5,.5)$ what is the probability that the resultant sum differs from the exact sum by more than $5$? [ANS]", "answer_v2": [ "0.00982327450749521" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$60$ numbers are rounded off to the nearest integer and then summed. If the individual round-off error are uniformly distributed over $(-.5,.5)$ what is the probability that the resultant sum differs from the exact sum by more than $4$? [ANS]", "answer_v3": [ "0.0736382701203026" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0332", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Central limit theorem", "level": "3", "keywords": [ "Probability", "Central Limit Theorem" ], "problem_v1": "A die is continuously rolled until the total sum of all rolls exceeds $350.$ What is the probability that at least $100$ rolls are necessary? [ANS]", "answer_v1": [ "0.570068" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A die is continuously rolled until the total sum of all rolls exceeds $150.$ What is the probability that at least $45$ rolls are necessary? [ANS]", "answer_v2": [ "0.345599" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A die is continuously rolled until the total sum of all rolls exceeds $225.$ What is the probability that at least $65$ rolls are necessary? [ANS]", "answer_v3": [ "0.514596" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Probability_0333", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Central limit theorem", "level": "3", "keywords": [ "Statistics" ], "problem_v1": "A variable of two populations has a mean of 48 and a standard deviation of 13 for one of the populations and a mean of 51 and a standard deviation of 14 for the other population.\na) For independent samples of sizes 39 and 40, respectively, find the mean and standard deviation of $\\overline{x}_1-\\overline{x}_2$. mean: [ANS]\nstandard deviation: [ANS]\nb) Can you conclude that the variable $\\overline{x}_1-\\overline{x}_2$ is approximately normally distributed? (Answer yes or no) answer: [ANS]", "answer_v1": [ "-3", "3.03864004668755", "YES" ], "answer_type_v1": [ "NV", "NV", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "A variable of two populations has a mean of 40 and a standard deviation of 10 for one of the populations and a mean of 55 and a standard deviation of 12 for the other population.\na) For independent samples of sizes 59 and 39, respectively, find the mean and standard deviation of $\\overline{x}_1-\\overline{x}_2$. mean: [ANS]\nstandard deviation: [ANS]\nb) Can you conclude that the variable $\\overline{x}_1-\\overline{x}_2$ is approximately normally distributed? (Answer yes or no) answer: [ANS]", "answer_v2": [ "-15", "2.32103919539179", "YES" ], "answer_type_v2": [ "NV", "NV", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "A variable of two populations has a mean of 43 and a standard deviation of 11 for one of the populations and a mean of 51 and a standard deviation of 13 for the other population.\na) For independent samples of sizes 36 and 40, respectively, find the mean and standard deviation of $\\overline{x}_1-\\overline{x}_2$. mean: [ANS]\nstandard deviation: [ANS]\nb) Can you conclude that the variable $\\overline{x}_1-\\overline{x}_2$ is approximately normally distributed? (Answer yes or no) answer: [ANS]", "answer_v3": [ "-8", "2.75428958374226", "YES" ], "answer_type_v3": [ "NV", "NV", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Probability_0334", "subject": "Probability", "topic": "Laws, theory", "subtopic": "Central limit theorem", "level": "4", "keywords": [ "statistics", "continuous random variables", "expectation" ], "problem_v1": "The wait time (after a scheduled arrival time) in minutes for a train to arrive is Uniformly distributed over the interval $[0, 15].$ You observe the wait time for the next $100$ trains to arrive. Assume wait times are independent. Part a) What is the approximate probability (to 2 decimal places) that the sum of the $100$ wait times you observed is between $706$ and $836$? [ANS]\nPart b) What is the approximate probability (to 2 decimal places) that the average of the $100$ wait times exceeds $7$ minutes? [ANS]\nPart c) Find the probability (to 2 decimal places) that $97$ or more of the $100$ wait times exceed $1$ minute. Please carry answers to at least 6 decimal places in intermediate steps. [ANS]\nPart d) Use the Normal approximation to the Binomial distribution (with continuity correction) to find the probability (to 2 decimal places) that $56$ or more of the $100$ wait times recorded exceed $5$ minutes. [ANS]", "answer_v1": [ "0.82", "0.88", "0.095", "0.99" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The wait time (after a scheduled arrival time) in minutes for a train to arrive is Uniformly distributed over the interval $[0, 12].$ You observe the wait time for the next $95$ trains to arrive. Assume wait times are independent. Part a) What is the approximate probability (to 2 decimal places) that the sum of the $95$ wait times you observed is between $536$ and $637$? [ANS]\nPart b) What is the approximate probability (to 2 decimal places) that the average of the $95$ wait times exceeds $6$ minutes? [ANS]\nPart c) Find the probability (to 2 decimal places) that $92$ or more of the $95$ wait times exceed $1$ minute. Please carry answers to at least 6 decimal places in intermediate steps. [ANS]\nPart d) Use the Normal approximation to the Binomial distribution (with continuity correction) to find the probability (to 2 decimal places) that $56$ or more of the $95$ wait times recorded exceed $5$ minutes. [ANS]", "answer_v2": [ "0.82", "0.5", "0.045", "0.49" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The wait time (after a scheduled arrival time) in minutes for a train to arrive is Uniformly distributed over the interval $[0, 12].$ You observe the wait time for the next $100$ trains to arrive. Assume wait times are independent. Part a) What is the approximate probability (to 2 decimal places) that the sum of the $100$ wait times you observed is between $565$ and $669$? [ANS]\nPart b) What is the approximate probability (to 2 decimal places) that the average of the $100$ wait times exceeds $6$ minutes? [ANS]\nPart c) Find the probability (to 2 decimal places) that $97$ or more of the $100$ wait times exceed $1$ minute. Please carry answers to at least 6 decimal places in intermediate steps. [ANS]\nPart d) Use the Normal approximation to the Binomial distribution (with continuity correction) to find the probability (to 2 decimal places) that $56$ or more of the $100$ wait times recorded exceed $5$ minutes. [ANS]", "answer_v3": [ "0.82", "0.5", "0.035", "0.72" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0335", "subject": "Probability", "topic": "Several variables", "subtopic": "Joint distribution", "level": "3", "keywords": [ "Joint Distribution", "Conditional", "Joint" ], "problem_v1": "The joint probability mass function of $X$ and $Y$ is given by \\begin{array}{lll} p(1,1)=0.3 & p(1,2)=0.1 & p(1,3)=0.1 \\cr p(2,1)=0.1 & p(2,2)=0.1 & p(2,3)=0.1 \\cr p(3,1)=0.05 & p(3,2)=0.05 & p(3,3)=0.1 \\end{array}\n(a) Compute the conditional mass function of $Y$ given $X=2$: $P(Y=1|X=2)=$ [ANS]\n$P(Y=2|X=2)=$ [ANS]\n$P(Y=3|X=2)=$ [ANS]\n(b) Are $X$ and $Y$ independent? (enter YES or NO) [ANS]\n(c) Compute the following probabilities: $P(X+Y>3)=$ [ANS]\n$P(XY=3)=$ [ANS]\n$P(\\frac{X}{Y} > 1)=$ [ANS]", "answer_v1": [ "0.333333333333333", "0.333333333333333", "0.333333333333333", "NO", "0.5", "0.15", "0.2" ], "answer_type_v1": [ "NV", "NV", "NV", "TF", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "The joint probability mass function of $X$ and $Y$ is given by \\begin{array}{lll} p(1,1)=0.1 & p(1,2)=0.05 & p(1,3)=0.1 \\cr p(2,1)=0.05 & p(2,2)=0.15 & p(2,3)=0.05 \\cr p(3,1)=0.1 & p(3,2)=0.05 & p(3,3)=0.35 \\end{array}\n(a) Compute the conditional mass function of $Y$ given $X=2$: $P(Y=1|X=2)=$ [ANS]\n$P(Y=2|X=2)=$ [ANS]\n$P(Y=3|X=2)=$ [ANS]\n(b) Are $X$ and $Y$ independent? (enter YES or NO) [ANS]\n(c) Compute the following probabilities: $P(X+Y>2)=$ [ANS]\n$P(XY=3)=$ [ANS]\n$P(\\frac{X}{Y} > 1)=$ [ANS]", "answer_v2": [ "0.2", "0.6", "0.2", "NO", "0.9", "0.2", "0.2" ], "answer_type_v2": [ "NV", "NV", "NV", "TF", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "The joint probability mass function of $X$ and $Y$ is given by \\begin{array}{lll} p(1,1)=0.45 & p(1,2)=0.05 & p(1,3)=0.1 \\cr p(2,1)=0.05 & p(2,2)=0.1 & p(2,3)=0.1 \\cr p(3,1)=0.05 & p(3,2)=0.05 & p(3,3)=0.05 \\end{array}\n(a) Compute the conditional mass function of $Y$ given $X=3$: $P(Y=1|X=3)=$ [ANS]\n$P(Y=2|X=3)=$ [ANS]\n$P(Y=3|X=3)=$ [ANS]\n(b) Are $X$ and $Y$ independent? (enter YES or NO) [ANS]\n(c) Compute the following probabilities: $P(X+Y>2)=$ [ANS]\n$P(XY=2)=$ [ANS]\n$P(\\frac{X}{Y} > 1)=$ [ANS]", "answer_v3": [ "0.333333333333333", "0.333333333333333", "0.333333333333333", "NO", "0.55", "0.1", "0.15" ], "answer_type_v3": [ "NV", "NV", "NV", "TF", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Probability_0336", "subject": "Probability", "topic": "Several variables", "subtopic": "Joint distribution", "level": "3", "keywords": [ "Joint Distribution", "Expected Value" ], "problem_v1": "The joint probability density function of $X$ and $Y$ is given by f(x,y)=c(y^2-256x^2)e^{-y}, \\ \\ \\-\\frac{y}{16} \\le x \\le \\frac{y}{16}, \\ \\ 0 < y < \\infty Find $c$ and the expected value of $X$: $c=$ [ANS]\n$E(X)=$ [ANS]", "answer_v1": [ "2", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The joint probability density function of $X$ and $Y$ is given by f(x,y)=c(y^2-4x^2)e^{-y}, \\ \\ \\-\\frac{y}{2} \\le x \\le \\frac{y}{2}, \\ \\ 0 < y < \\infty Find $c$ and the expected value of $X$: $c=$ [ANS]\n$E(X)=$ [ANS]", "answer_v2": [ "0.25", "0" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The joint probability density function of $X$ and $Y$ is given by f(x,y)=c(y^2-64x^2)e^{-y}, \\ \\ \\-\\frac{y}{8} \\le x \\le \\frac{y}{8}, \\ \\ 0 < y < \\infty Find $c$ and the expected value of $X$: $c=$ [ANS]\n$E(X)=$ [ANS]", "answer_v3": [ "1", "0" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Probability_0337", "subject": "Probability", "topic": "Several variables", "subtopic": "Joint distribution", "level": "3", "keywords": [ "Joint Distribution", "Expected Value", "Independant" ], "problem_v1": "Let f(x)=\\left\\{\\begin{array}{ll} c x^{8} y^{7} & \\text{if}\\ 0 \\le x \\le 1, \\ 0 \\le y \\le 1 \\cr 0 & \\mbox{otherwise} \\end{array} \\right. Find the following:\n(a) $c$ such that $f(x,y)$ is a probability density function: $c=$ [ANS]\n(b) Expected values of $X$ and $Y$: $E(X)=$ [ANS]\n$E(Y)=$ [ANS]\n(c) Are $X$ and $Y$ independent? (enter YES or NO) [ANS]", "answer_v1": [ "72", "0.9", "0.888888888888889", "YES" ], "answer_type_v1": [ "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let f(x)=\\left\\{\\begin{array}{ll} c x^{2} y^{10} & \\text{if}\\ 0 \\le x \\le 1, \\ 0 \\le y \\le 1 \\cr 0 & \\mbox{otherwise} \\end{array} \\right. Find the following:\n(a) $c$ such that $f(x,y)$ is a probability density function: $c=$ [ANS]\n(b) Expected values of $X$ and $Y$: $E(X)=$ [ANS]\n$E(Y)=$ [ANS]\n(c) Are $X$ and $Y$ independent? (enter YES or NO) [ANS]", "answer_v2": [ "33", "0.75", "0.916666666666667", "YES" ], "answer_type_v2": [ "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let f(x)=\\left\\{\\begin{array}{ll} c x^{4} y^{7} & \\text{if}\\ 0 \\le x \\le 1, \\ 0 \\le y \\le 1 \\cr 0 & \\mbox{otherwise} \\end{array} \\right. Find the following:\n(a) $c$ such that $f(x,y)$ is a probability density function: $c=$ [ANS]\n(b) Expected values of $X$ and $Y$: $E(X)=$ [ANS]\n$E(Y)=$ [ANS]\n(c) Are $X$ and $Y$ independent? (enter YES or NO) [ANS]", "answer_v3": [ "40", "0.833333333333333", "0.888888888888889", "YES" ], "answer_type_v3": [ "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Probability_0338", "subject": "Probability", "topic": "Several variables", "subtopic": "Marginal distributions", "level": "2", "keywords": [ "joint", "probability", "marginal", "mean", "variance" ], "problem_v1": "The bivariate distribution of X and Y is described below:\n$\\begin{array}{ccc}\\hline & X & \\\\ \\hline Y & 1 & 2 \\\\ \\hline 1 & 0.27 & 0.46 \\\\ \\hline 2 & 0.12 & 0.15 \\\\ \\hline \\end{array}$\nA. Find the marginal probability distribution of X. 1: [ANS]\n2: [ANS]\nB. Find the marginal probability distribution of Y. 1: [ANS]\n2: [ANS]\nC. Compute the mean and variance of X.\nMean=[ANS]\nVariance=[ANS]\nC. Compute the mean and variance of Y.\nMean=[ANS]\nVariance=[ANS]", "answer_v1": [ "0.39", "0.61", "0.73", "0.27", "1.61", "0.2379", "1.27", "0.1971" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "The bivariate distribution of X and Y is described below:\n$\\begin{array}{ccc}\\hline & X & \\\\ \\hline Y & 1 & 2 \\\\ \\hline 1 & 0.2 & 0.41 \\\\ \\hline 2 & 0.14 & 0.25 \\\\ \\hline \\end{array}$\nA. Find the marginal probability distribution of X. 1: [ANS]\n2: [ANS]\nB. Find the marginal probability distribution of Y. 1: [ANS]\n2: [ANS]\nC. Compute the mean and variance of X.\nMean=[ANS]\nVariance=[ANS]\nC. Compute the mean and variance of Y.\nMean=[ANS]\nVariance=[ANS]", "answer_v2": [ "0.34", "0.66", "0.61", "0.39", "1.66", "0.2244", "1.39", "0.2379" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "The bivariate distribution of X and Y is described below:\n$\\begin{array}{ccc}\\hline & X & \\\\ \\hline Y & 1 & 2 \\\\ \\hline 1 & 0.23 & 0.42 \\\\ \\hline 2 & 0.13 & 0.22 \\\\ \\hline \\end{array}$\nA. Find the marginal probability distribution of X. 1: [ANS]\n2: [ANS]\nB. Find the marginal probability distribution of Y. 1: [ANS]\n2: [ANS]\nC. Compute the mean and variance of X.\nMean=[ANS]\nVariance=[ANS]\nC. Compute the mean and variance of Y.\nMean=[ANS]\nVariance=[ANS]", "answer_v3": [ "0.36", "0.64", "0.65", "0.35", "1.64", "0.2304", "1.35", "0.2275" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] } ]