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Some Important Discrete Probability Distributions 123 CHAPTER 5: SOME IMPORTANT DISCRETE PROBABILITY DISTRIBUTIONS 1. Thirty-six of the staff of 80 teachers at a local intermediate school are certified in CardioPulmonary Resuscitation (CPR). In 180 days of school, about how many days can we expect that the teacher on bus duty will likely be certified in CPR? a) 5 days b) 45 days c) 65 days d) 81 days ANSWER: d TYPE: MC DIFFICULTY: Moderate KEYWORDS: binomial distribution, mean 2. A campus program evenly enrolls undergraduate and graduate students. If a random sample of 4 students is selected from the program to be interviewed about the introduction of a new fast food outlet on the ground floor of the campus building, what is the probability that all 4 students selected are undergraduate students? a) 0.0256 b) 0.0625 c) 0.16 d) 1.00 ANSWER: b TYPE: MC DIFFICULTY: Difficult KEYWORDS: binomial distribution 3. A probability distribution is an equation that a) associates a particular probability of occurrence with each outcome in the sample space. b) measures outcomes and assigns values of X to the simple events. c) assigns a value to the variability in the sample space. d) assigns a value to the center of the sample space. ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: probability distribution 4. The connotation "expected value" or "expected gain" from playing roulette at a casino means a) the amount you expect to "gain" on a single play. b) the amount you expect to "gain" in the long run over many plays. c) the amount you need to "break even" over many plays. d) the amount you should expect to gain if you are lucky. ANSWER: b 124 Some Important Discrete Probability Distributions TYPE: MC DIFFICULTY: Easy KEYWORDS: expected value 5. Which of the following about the binomial distribution is not a true statement? a) The probability of success must be constant from trial to trial. b) Each outcome is independent of the other. c) Each outcome may be classified as either "success" or "failure." d) The random variable of interest is continuous. ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: binomial distribution, properties 6. In a binomial distribution a) the random variable X is continuous. b) the probability of success p is stable from trial to trial. c) the number of trials n must be at least 30. d) the results of one trial are dependent on the results of the other trials. ANSWER: b TYPE: MC DIFFICULTY: Easy KEYWORDS: binomial distribution, properties 7. Whenever p = 0.5, the binomial distribution will a) always be symmetric. b) be symmetric only if n is large. c) be right-skewed. d) be left-skewed. ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: binomial distribution, properties 8. Whenever p = 0.1 and n is small, the binomial distribution will be a) symmetric. b) right-skewed. c) left-skewed. d) None of the above. ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: binomial distribution, properties 9. If n = 10 and p = 0.70, then the mean of the binomial distribution is a) 0.07. b) 1.45. c) 7.00. Some Important Discrete Probability Distributions 125 d) 14.29. ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: binomial distribution, mean 10. If n = 10 and p = 0.70, then the standard deviation of the binomial distribution is a) 0.07. b) 1.45. c) 7.00. d) 14.29. ANSWER: b TYPE: MC DIFFICULTY: Easy KEYWORDS: binomial distribution, standard deviation 11. If the outcomes of a random variable follow a Poisson distribution, then their a) mean equals the standard deviation. b) median equals the standard deviation. c) mean equals the variance. d) median equals the variance. ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: Poisson distribution, mean, standard deviation, properties 12. What type of probability distribution will the consulting firm most likely employ to analyze the insurance claims in the following problem? An insurance company has called a consulting firm to determine if the company has an unusually high number of false insurance claims. It is known that the industry proportion for false claims is 3%. The consulting firm has decided to randomly and independently sample 100 of the company’s insurance claims. They believe the number of these 100 that are false will yield the information the company desires. a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: a TYPE: MC DIFFICULTY: Difficult KEYWORDS: binomial distribution, properties 13. The covariance a) must be between -1 and +1. b) must be positive. c) can be positive or negative. d) must be less than +1. 126 Some Important Discrete Probability Distributions ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: covariance, properties 14. What type of probability distribution will most likely be used to analyze warranty repair needs on new cars in the following problem? The service manager for a new automobile dealership reviewed dealership records of the past 20 sales of new cars to determine the number of warranty repairs he will be called on to perform in the next 90 days. Corporate reports indicate that the probability any one of their new cars needs a warranty repair in the first 90 days is 0.05. The manager assumes that calls for warranty repair are independent of one another and is interested in predicting the number of warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new cars sold. a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: a TYPE: MC DIFFICULTY: Difficult KEYWORDS: binomial distribution, properties 15. What type of probability distribution will most likely be used to analyze the number of chocolate chip parts per cookie in the following problem? The quality control manager of Marilyn’s Cookies is inspecting a batch of chocolate chip cookies. When the production process is in control, the average number of chocolate chip parts per cookie is 6.0. The manager is interested in analyzing the probability that any particular cookie being inspected has fewer than 5.0 chip parts. a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: Poisson distribution, properties 16. What type of probability distribution will most likely be used to analyze the number of cars with defective radios in the following problem? From an inventory of 48 new cars being shipped to local dealerships, corporate reports indicate that 12 have defective radios installed. The sales manager of one dealership wants to predict the probability out of the 8 new cars it just received that, when each is tested, no more than 2 of the cars have defective radios. a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. Some Important Discrete Probability Distributions 127 ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution, properties 17. A stock analyst was provided with a list of 25 stocks. He was expected to pick 3 stocks from the list whose prices are expected to rise by more than 20% after 30 days. In reality, the prices of only 5 stocks would rise by more than 20% after 30 days. If he randomly selected 3 stocks from the list, he would use what type of probability distribution to compute the probability that all of the chosen stocks would appreciate more than 20% after 30 days? a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution, properties 18. A professor receives, on average, 24.7 e-mails from students the day before the midterm exam. To compute the probability of receiving at least 10 e-mails on such a day, he will use what type of probability distribution? a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: Poisson distribution, properties 19. A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.025. To find the probability that exactly 20 of the computers will require repair on a given day, one will use what type of probability distribution? a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: binomial distribution, properties 20. The portfolio expected return of two investments a) will be higher when the covariance is zero. b) will be higher when the covariance is negative. c) will be higher when the covariance is positive. 128 Some Important Discrete Probability Distributions d) does not depend on the covariance. ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: portfolio, mean 21. A financial analyst is presented with information on the past records of 60 start-up companies and told that in fact only 3 of them have managed to become highly successful. He selected 3 companies from this group as the candidates for success. To analyze his ability to spot the companies that will eventually become highly successful, he will use what type of probability distribution? a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: c TYPE: MC DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution, properties 22. On the average, 1.8 customers per minute arrive at any one of the checkout counters of a grocery store. What type of probability distribution can be used to find out the probability that there will be no customer arriving at a checkout counter? a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: Poisson distribution, properties 23. A multiple-choice test has 30 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all questions randomly. What type of probability distribution can be used to figure out his chance of getting at least 20 questions right? a) binomial distribution b) Poisson distribution c) hypergeometric distribution d) None of the above. ANSWER: a TYPE: MC DIFFICULTY: Moderate KEYWORDS: binomial distribution, properties Some Important Discrete Probability Distributions 129 24. A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Suppose the mean cost of rats used in lab experiments turned out to be $13.00 per week. Interpret this value. a) Most of the weeks resulted in rat costs of $13.00. b) The median cost for the distribution of rat costs is $13.00. c) The expected or average cost for all weekly rat purchases is $13.00. d) The rat cost that occurs more often than any other is $13.00. ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: mean 25. A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Prices for 100 rats follow the following distribution: Price: $10.00 $12.50 $15.00 Probability: 0.35 0.40 0.25 How much should the lab budget for next year’s rat orders be, assuming this distribution does not change? a) $520 b) $637 c) $650 d) $780 ANSWER: b TYPE: MC DIFFICULTY: Moderate KEYWORDS: mean 26. The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day. Interpret the value of the mean. a) The number of tickets that is written most often is 6.5 tickets per day. b) Half of the days have less than 6.5 tickets written and half of the days have more than 6.5 tickets written. c) If we sampled all days, the arithmetic average or expected number of tickets written would be 6.5 tickets per day. d) The mean has no interpretation since 0.5 ticket can never be written. ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: mean, Poisson distribution 27. True or False: The Poisson distribution can be used to model a continuous random variable. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: Poisson distribution, properties 130 Some Important Discrete Probability Distributions 28. True or False: Another name for the mean of a probability distribution is its expected value. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: mean 29. True or False: The number of customers arriving at a department store in a 5-minute period has a binomial distribution. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: Poisson distribution, properties 30. True or False: The number of customers arriving at a department store in a 5-minute period has a Poisson distribution. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: Poisson distribution, properties 31. True or False: The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a binomial distribution. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: hypergeometric distribution, properties 32. True or False: The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a hypergeometric distribution. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: hypergeometric distribution, properties 33. True or False: The diameters of 10 randomly selected bolts have a binomial distribution. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: binomial distribution, properties 34. True or False: The largest value that a Poisson random variable X can have is n. ANSWER: False TYPE: TF DIFFICULTY: Moderate Some Important Discrete Probability Distributions 131 KEYWORDS: Poisson distribution, properties 35. True or False: In a Poisson distribution, the mean and standard deviation are equal. ANSWER: False TYPE: TF DIFFICULTY: Moderate KEYWORDS: Poisson distribution, properties 36. True or False: In a Poisson distribution, the mean and variance are equal. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: Poisson distribution, properties 37. True or False: If p remains constant in a binomial distribution, an increase in n will increase the variance. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: binomial distribution, properties 38. True or False: If p remains constant in a binomial distribution, an increase in n will not change the mean. ANSWER: False TYPE: TF DIFFICULTY: Moderate KEYWORDS: binomial distribution, properties 39. True or False: Suppose that a judge’s decisions follow a binomial distribution and that his verdict is correct 90% of the time. In his next 10 decisions, the probability that he makes fewer than 2 incorrect verdicts is 0.736. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: binomial distribution, probability 40. True or False: Suppose that the number of airplanes arriving at an airport per minute is a Poisson process. The average number of airplanes arriving per minute is 3. The probability that exactly 6 planes arrive in the next minute is 0.0504. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: Poisson distribution, probability 132 Some Important Discrete Probability Distributions 41. True or False: The covariance between two investments is equal to the sum of the variances of the investments. ANSWER: False TYPE: TF DIFFICULTY: Easy KEYWORDS: covariance, variance 42. True or False: If the covariance between two investments is zero, the variance of the sum of the two investments will be equal to the sum of the variances of the investments. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: covariance, variance 43. True or False: The expected return of the sum of two investments will be equal to the sum of the expected returns of the two investments plus twice the covariance between the investments. ANSWER: False TYPE: TF DIFFICULTY: Moderate KEYWORDS: mean, portfolio 44. True or False: The variance of the sum of two investments will be equal to the sum of the variances of the two investments plus twice the covariance between the investments. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: covariance, variance 45. True or False: The variance of the sum of two investments will be equal to the sum of the variances of the two investments when the covariance between the investments is zero. ANSWER: True TYPE: TF DIFFICULTY: Moderate KEYWORDS: covariance, variance 46. True or False: The expected return of a two-asset portfolio is equal to the product of the weight assigned to the first asset and the expected return of the first asset plus the product of the weight assigned to the second asset and the expected return of the second asset. ANSWER: True TYPE: TF DIFFICULTY: Easy KEYWORDS: mean, portfolio TABLE 5-1 Some Important Discrete Probability Distributions 133 The probability that a particular type of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. You have 2 such alarms in your home and they operate independently. 47. Referring to Table 5-1, the probability that both sound an alarm in the presence of smoke is ________. ANSWER: 0.64 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 48. Referring to Table 5-1, the probability that neither sound an alarm in the presence of smoke is ________. ANSWER: 0.04 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 49. Referring to Table 5-1, the probability that at least one sounds an alarm in the presence of smoke is ________. ANSWER: 0.96 TYPE: FI DIFFICULTY: Difficult KEYWORDS: binomial distribution TABLE 5-2 A certain type of new business succeeds 60% of the time. Suppose that 3 such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent). 50. Referring to Table 5-2, the probability that all 3 businesses succeed is ________. ANSWER: 0.216 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 51. Referring to Table 5-2, the probability that all 3 businesses fail is ________. ANSWER: 0.064 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 52. Referring to Table 5-2, the probability that at least 1 business succeeds is ________. ANSWER: 0.936 134 Some Important Discrete Probability Distributions TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 53. Referring to Table 5-2, the probability that exactly 1 business succeeds is ________. ANSWER: 0.288 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 54. If X has a binomial distribution with n = 4 and p = 0.3, then P(X = 1) = ________ . ANSWER: 0.4116 TYPE: FI DIFFICULTY: Easy KEYWORDS: binomial distribution 55. If X has a binomial distribution with n = 4 and p = 0.3, then P(X > 1) = ________ . ANSWER: 0.3483 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 56. If X has a binomial distribution with n = 5 and p = 0.1, then P(X = 2) = ________ . ANSWER: 0.0729 TYPE: FI DIFFICULTY: Easy KEYWORDS: binomial distribution 57. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The probability that exactly 1 prefers brand C is ________. ANSWER: 0.0768 TYPE: FI DIFFICULTY: Easy KEYWORDS: binomial distribution 58. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The probability that at least 1 prefers brand C is ________. ANSWER: 0.9898 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 59. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The probability that exactly 3 prefer brand C is ________. ANSWER: Some Important Discrete Probability Distributions 135 0.3456 TYPE: FI DIFFICULTY: Easy KEYWORDS: binomial distribution 60. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The probability that exactly 4 prefer brand C is ________. ANSWER: 0.2592 TYPE: FI DIFFICULTY: Easy KEYWORDS: binomial distribution 61. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The probability that at most 2 prefer brand C is ________. ANSWER: 0.3174 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 62. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The probability that more than 3 prefer brand C is ________. ANSWER: 0.3370 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 63. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The probability that less than 2 prefer brand C is ________. ANSWER: 0.0870 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 64. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The average number that you would expect to prefer brand C is ________. ANSWER: 3 TYPE: FI DIFFICULTY: Easy KEYWORDS: binomial distribution, mean 65. Suppose that past history shows that 60% of college students prefer Brand C cola. A sample of 5 students is to be selected. The variance of the number that prefer brand C is ________. ANSWER: 1.2 TYPE: FI DIFFICULTY: Easy 136 Some Important Discrete Probability Distributions KEYWORDS: binomial distribution, variance TABLE 5-3 The following table contains the probability distribution for X = the number of retransmissions necessary to successfully transmit a 1024K data package through a double satellite media. X 0 1 2 3 P(X) 0.35 0.35 0.25 0.05 66. Referring to Table 5-3, the probability of no retransmissions is ________. ANSWER: 0.35 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution 67. Referring to Table 5-3, the probability of at least one retransmission is ________. ANSWER: 0.65 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution 68. Referring to Table 5-3, the mean or expected value for the number of retransmissions is ________. ANSWER: 1.0 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution, mean 69. Referring to Table 5-3, the variance for the number of retransmissions is ________. ANSWER: 0.80 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution, variance 70. Referring to Table 5-3, the standard deviation of the number of retransmissions is ________. ANSWER: 0.894 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution, standard deviation 71. In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The probability that she does not get audited is ________. ANSWER: Some Important Discrete Probability Distributions 137 0.2373 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 72. In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The probability that she gets audited once is ________. ANSWER: 0.3955 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 73. In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The probability that she gets audited at least once is ________. ANSWER: 0.7627 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 74. In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The probability that she gets audited no more than 2 times is ________. ANSWER: 0.8965 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution 75. In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The expected number of times she will be audited is ________. ANSWER: 1.25 TYPE: FI DIFFICULTY: Easy KEYWORDS: binomial distribution, mean 76. In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The variance of the number of times she will be audited is ________. ANSWER: 0.9375 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution, variance 138 Some Important Discrete Probability Distributions 77. In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The standard deviation of the number of times she will be audited is ________. ANSWER: 0.968 TYPE: FI DIFFICULTY: Moderate KEYWORDS: binomial distribution, standard deviation TABLE 5-4 The following table contains the probability distribution for X = the number of traffic accidents reported in a day in Corvallis, Oregon. X 0 1 2 3 4 5 P(X) 0.10 0.20 0.45 0.15 0.05 0.05 78. Referring to Table 5-4, the probability of 3 accidents is ________. ANSWER: 0.15 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution 79. Referring to Table 5-4, the probability of at least 1 accident is ________. ANSWER: 0.90 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution 80. Referring to Table 5-4, the mean or expected value of the number of accidents is ________. ANSWER: 2.0 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution, mean, 81. Referring to Table 5-4, the variance of the number of accidents is ________. ANSWER: 1.4 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution, variance 82. Referring to Table 5-4, the standard deviation of the number of accidents is ________. ANSWER: 1.18 TYPE: FI DIFFICULTY: Easy KEYWORDS: probability distribution, standard deviation Some Important Discrete Probability Distributions 139 83. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The probability that there will be exactly 3 power outages in a year is ____________. ANSWER: 0.0892 TYPE: FI DIFFICULTY: Easy KEYWORDS: Poisson distribution 84. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The probability that there will be at least 3 power outages in a year is ____________. ANSWER: 0.9380 TYPE: FI DIFFICULTY: Difficult KEYWORDS: Poisson distribution 85. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The probability that there will be at least 1 power outage in a year is ____________. ANSWER: 0.9975 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Poisson distribution 86. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The probability that there will be no more than 1 power outage in a year is ____________. ANSWER: 0.0174 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Poisson distribution 87. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The probability that there will be between 1 and 3 inclusive power outages in a year is ____________. ANSWER: 0.1487 TYPE: FI DIFFICULTY: Difficult KEYWORDS: Poisson distribution 88. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The variance of the number of power outages is ____________. ANSWER: 6 140 Some Important Discrete Probability Distributions TYPE: FI DIFFICULTY: Easy KEYWORDS: Poisson distribution 89. The number of 911 calls in Butte, Montana, has a Poisson distribution with a mean of 10 calls a day. The probability of seven 911 calls in a day is ____________. ANSWER: 0.0901 TYPE: FI DIFFICULTY: Easy KEYWORDS: Poisson distribution 90. The number of 911 calls in Butte, Montana, has a Poisson distribution with a mean of 10 calls a day. The probability of seven or eight 911 calls in a day is ____________. ANSWER: 0.2027 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Poisson distribution 91. The number of 911 calls in Butte, Montana, has a Poisson distribution with a mean of 10 calls a day. The probability of 2 or more 911 calls in a day is ____________. ANSWER: 0.9995 TYPE: FI DIFFICULTY: Moderate KEYWORDS: Poisson distribution 92. The number of 911 calls in Butte, Montana, has a Poisson distribution with a mean of 10.0 calls a day. The standard deviation of the number of 911 calls in a day is ____________ . ANSWER: 3.16 TYPE: FI DIFFICULTY: Easy KEYWORDS: Poisson distribution 93. An Undergraduate Study Committee of 6 members at a major university is to be formed from a pool of faculty of 18 men and 6 women. If the committee members are chosen randomly, what is the probability that precisely half of the members will be women? ANSWER: 0.1213 TYPE: FI DIFFICULTY: Easy KEYWORDS: hypergeometric distribution 94. An Undergraduate Study Committee of 6 members at a major university is to be formed from a pool of faculty of 18 men and 6 women. If the committee members are chosen randomly, what is the probability that all of the members will be men? ANSWER: 0.1379 Some Important Discrete Probability Distributions 141 TYPE: FI DIFFICULTY: Easy KEYWORDS: hypergeometric distribution 95. A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students. Ten of the 15 students have no prior competition experience while the others have some degree of experience. What is the probability that none of the members chosen for the team have any competition experience? ANSWER: 0.1538 TYPE: FI DIFFICULTY: Easy KEYWORDS: hypergeometric distribution 96. A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students. Ten of the 15 students have no prior competition experience while the others have some degree of experience. What is the probability that at least 1 of the members chosen for the team have some prior competition experience? ANSWER: 0.8462 TYPE: FI DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution 97. A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students. Ten of the 15 students have no prior competition experience while the others have some degree of experience. What is the probability that at most 1 of the members chosen for the team have some prior competition experience? ANSWER: 0.5934 TYPE: FI DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution 98. A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students. Ten of the 15 students have no prior competition experience while the others have some degree of experience. What is the probability that exactly half of the members chosen for the team have some prior competition experience? ANSWER: 0.3297 TYPE: FI DIFFICULTY: Easy KEYWORDS: hypergeometric distribution 99. The Department of Commerce in a particular state has determined that the number of small businesses that declare bankruptcy per month is approximately a Poisson distribution with a mean of 6.4. Find the probability that more than 3 bankruptcies occur next month. ANSWER: 0.881 TYPE: PR DIFFICULTY: Moderate KEYWORDS: Poisson distribution 142 Some Important Discrete Probability Distributions 100. The Department of Commerce in a particular state has determined that the number of small businesses that declare bankruptcy per month is approximately a Poisson distribution with a mean of 6.4. Find the probability that exactly 5 bankruptcies occur next month. ANSWER: 0.149 TYPE: PR DIFFICULTY: Easy KEYWORDS: Poisson distribution 101. The on-line access computer service industry is growing at an extraordinary rate. Current estimates suggest that only 20% of the home-based computers have access to on-line services. This number is expected to grow quickly over the next 5 years. Suppose 25 people with homebased computers were randomly and independently sampled. Find the probability that fewer than 10 of those sampled currently have access to on-line services. ANSWER: 0.983 TYPE: PR DIFFICULTY: Moderate KEYWORDS: binomial distribution 102. The on-line access computer service industry is growing at an extraordinary rate. Current estimates suggest that only 20% of the home-based computers have access to on-line services. This number is expected to grow quickly over the next 5 years. Suppose 25 people with homebased computers were randomly and independently sampled. Find the probability that more than 20 of those sampled currently do not have access to on-line services. ANSWER: 0.421 TYPE: PR DIFFICULTY: Difficult KEYWORDS: binomial distribution 103. A national trend predicts that women will account for half of all business travelers in the next 3 years. To attract these women business travelers, hotels are providing more amenities that women particularly like. A recent survey of American hotels found that 70% offer hairdryers in the bathrooms. Consider a random and independent sample of 20 hotels. Find the probability all of the hotels in the sample offered hairdryers in the bathrooms. ANSWER: 0.0008 TYPE: PR DIFFICULTY: Moderate KEYWORDS: binomial distribution 104. A national trend predicts that women will account for half of all business travelers in the next 3 years. To attract these women business travelers, hotels are providing more amenities that women particularly like. A recent survey of American hotels found that 70% offer hairdryers in the bathrooms. Consider a random and independent sample of 20 hotels. Find the probability that more than 7 but less than 13 of the hotels in the sample offered hairdryers in the bathrooms. ANSWER: 0.2264 Some Important Discrete Probability Distributions 143 TYPE: PR DIFFICULTY: Moderate KEYWORDS: binomial distribution 105. A national trend predicts that women will account for half of all business travelers in the next 3 years. To attract these women business travelers, hotels are providing more amenities that women particularly like. A recent survey of American hotels found that 70% offer hairdryers in the bathrooms. Consider a random and independent sample of 20 hotels. Find the probability that at least 9 of the hotels in the sample do not offer hairdryers in the bathrooms. ANSWER: 0.1133 TYPE: PR DIFFICULTY: Difficult KEYWORDS: binomial distribution 106. The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.4 tickets per day. Find the probability that less than 6 tickets are written on a randomly selected day from this population. ANSWER: 0.384 TYPE: PR DIFFICULTY: Easy KEYWORDS: Poisson distribution 107. The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.4 tickets per day. Find the probability that exactly 6 tickets are written on a randomly selected day from this population. ANSWER: 0.159 TYPE: PR DIFFICULTY: Easy KEYWORDS: Poisson distribution TABLE 5-5 From an inventory of 48 new cars being shipped to local dealerships, corporate reports indicate that 12 have defective radios installed. 108. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, no more than 2 of the cars have defective radios? ANSWER: 0.6863 TYPE: PR DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution 109. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, exactly half of the cars have defective radios? ANSWER: 144 Some Important Discrete Probability Distributions 0.0773 TYPE: PR DIFFICULTY: Easy KEYWORDS: hypergeometric distribution 110. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, none of the cars have defective radios? ANSWER: 0.08019 TYPE: PR DIFFICULTY: Easy KEYWORDS: hypergeometric distribution 111. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, at least half of the cars have defective radios? ANSWER: 0.09388 TYPE: PR DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution 112. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, no more than half of the cars have defective radios? ANSWER: 0.9834 TYPE: PR DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution 113. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, at most 2 of the cars have defective radios? ANSWER: 0.6863 TYPE: PR DIFFICULTY: Moderate KEYWORDS: hypergeometric distribution 114. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, exactly two of the cars have non-defective radios? ANSWER: 0.001543 TYPE: PR DIFFICULTY: Difficult KEYWORDS: hypergeometric distribution 115. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, at most three of the cars have non-defective radios? ANSWER: 0.01661 TYPE: PR DIFFICULTY: Difficult KEYWORDS: hypergeometric distribution Some Important Discrete Probability Distributions 145 116. Referring to Table 5-5, what is the probability out of the 8 new cars it just received that, when each is tested, no more than half of the cars have non-defective radios? ANSWER: 0.09388 TYPE: PR DIFFICULTY: Difficult KEYWORDS: hypergeometric distribution TABLE 5-6 The quality control manager of Marilyn’s Cookies is inspecting a batch of chocolate chip cookies. When the production process is in control, the average number of chocolate chip parts per cookie is 6.0. 117. Referring to Table 5-6, what is the probability that any particular cookie being inspected has 4.0 chip parts. ANSWER: 0.1339 TYPE: PR DIFFICULTY: Easy KEYWORDS: Poisson distribution 118. Referring to Table 5-6, what is the probability that any particular cookie being inspected has fewer than 5.0 chip parts. ANSWER: 0.2851 TYPE: PR DIFFICULTY: Moderate KEYWORDS: Poisson distribution 119. Referring to Table 5-6, what is the probability that any particular cookie being inspected has at least 6.0 chip parts. ANSWER: 0.5543 TYPE: PR DIFFICULTY: Moderate KEYWORDS: Poisson distribution 120. Referring to Table 5-6, what is the probability that any particular cookie being inspected has between 5.0 and 8.0 inclusive chip parts. ANSWER: 0.5622 TYPE: PR DIFFICULTY: Moderate KEYWORDS: Poisson distribution 121. Referring to Table 5-6, what is the probability that any particular cookie being inspected has less than 5.0 or more than 8.0 chip parts. ANSWER: 146 Some Important Discrete Probability Distributions 0.4378 TYPE: PR DIFFICULTY: Difficult KEYWORDS: Poisson distribution TABLE 5-7 There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution: Probability .25 .40 .35 Returns Neighborhood A Neighborhood B $22,500 $30,500 $10,000 $25,000 $40,500 $10,500 122. Referring to Table 5-7, what is the expected value gain for the house in neighborhood A? ANSWER: $ 12,550 TYPE: PR DIFFICULTY: Easy KEYWORDS: mean 123. Referring to Table 5-7, what is the expected value gain for the house in neighborhood B? ANSWER: $ 21,300 TYPE: PR DIFFICULTY: Easy KEYWORDS: mean 124. Referring to Table 5-7, what is the variance of the gain in value for the house in neighborhood A? ANSWER: 583,147,500 TYPE: PR DIFFICULTY: Easy KEYWORDS: variance 125. Referring to Table 5-7, what is the variance of the gain in value for the house in neighborhood B? ANSWER: 67,460,000 TYPE: PR DIFFICULTY: Easy KEYWORDS: variance 126. Referring to Table 5-7, what is the standard deviation of the value gain for the house in neighborhood A? ANSWER: Some Important Discrete Probability Distributions 147 $24,148.45 TYPE: PR DIFFICULTY: Easy KEYWORDS: standard deviation 127. Referring to Table 5-7, what is the standard deviation of the value gain for the house in neighborhood B? ANSWER: $8,213.40 TYPE: PR DIFFICULTY: Easy KEYWORDS: standard deviation 128. Referring to Table 5-7, what is the covariance of the two houses? ANSWER: 190,040,000 TYPE: PR DIFFICULTY: Easy KEYWORDS: covariance 129. Referring to Table 5-7, what is the expected value gain if you invest in both houses? ANSWER: $33,850 TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean of the sum 130. Referring to Table 5-7, what is the total variance of value gain if you invest in both houses? ANSWER: 270,527,500 TYPE: PR DIFFICULTY: Moderate KEYWORDS: variance of the sum 131. Referring to Table 5-7, what is the total standard deviation of value gain if you invest in both houses? ANSWER: $16,447.72 TYPE: PR DIFFICULTY: Moderate KEYWORDS: standard deviation of the sum 132. Referring to Table 5-7, if you can invest half of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio expected return of your investment? ANSWER: $16,925 TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, portfolio 148 Some Important Discrete Probability Distributions 133. Referring to Table 5-7, if you can invest half of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio risk of your investment? ANSWER: $8,223.86 TYPE: PR DIFFICULTY: Moderate KEYWORDS: standard deviation, risk, portfolio 134. Referring to Table 5-7, if you can invest 10% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio expected return of your investment? ANSWER: $20,425 TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, portfolio 135. Referring to Table 5-7, if you can invest 10% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio risk of your investment? ANSWER: $5,125.12 TYPE: PR DIFFICULTY: Moderate KEYWORDS: standard deviation, risk, portfolio 136. Referring to Table 5-7, if you can invest 30% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio expected return of your investment? ANSWER: $18,675 TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, portfolio 137. Referring to Table 5-7, if you can invest 30% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio risk of your investment? ANSWER: $2,392.04 TYPE: PR DIFFICULTY: Moderate KEYWORDS: standard deviation, risk, portfolio 138. Referring to Table 5-7, if you can invest 70% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio expected return of your investment? ANSWER: $15,175 TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, portfolio Some Important Discrete Probability Distributions 149 139. Referring to Table 5-7, if you can invest 70% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio risk of your investment? ANSWER: $14,560.11 TYPE: PR DIFFICULTY: Moderate KEYWORDS: standard deviation, risk, portfolio 140. Referring to Table 5-7, if you can invest 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio expected return of your investment? ANSWER: $13,425 TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, portfolio 141. Referring to Table 5-7, if you can invest 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio risk of your investment? ANSWER: $20,947.96 TYPE: PR DIFFICULTY: Moderate KEYWORDS: standard deviation, risk, portfolio 142. Referring to Table 5-7, if your investment preference is to maximize your expected return while exposing yourself to the minimal amount of risk, will you choose a portfolio that will consist of 10%, 30%, 50%, 70%, or 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B? ANSWER: 30% TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, standard deviation, risk, portfolio, coefficient of variation 143. Referring to Table 5-7, if your investment preference is to maximize your expected return and not worry at all about the risk that you have to take, will you choose a portfolio that will consist of 10%, 30%, 50%, 70%, or 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B? ANSWER: 10% TYPE: PR DIFFICULTY: Moderate KEYWORDS: mean, portfolio 144. Referring to Table 5-7, if your investment preference is to minimize the amount of risk that you have to take and do not care at all about the expected return, will you choose a portfolio that will consist of 10%, 30%, 50%, 70%, or 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B? ANSWER: 150 Some Important Discrete Probability Distributions 30% TYPE: PR DIFFICULTY: Moderate KEYWORDS: standard deviation, risk, portfolio