Discrete Probability Distributions 5-1 CHAPTER 5: DISCRETE PROBABILITY DISTRIBUTIONS 1. Thirty-six of the staff of 80 teachers at a local intermediate school are certified in Cardio- Pulmonary 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 d TYPE: MC DIFFICULTY: Moderate, 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 b TYPE: MC DIFFICULTY: Difficult 3. A probability distribution is an equation that a) associates a particular probability of occurrence with each outcome. b) measures outcomes and assigns values of X to the simple events. c) assigns a value to the variability of the set of events. d) assigns a value to the center of the set of events. 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. b TYPE: MC DIFFICULTY: Easy KEYWORDS: expected value
5-2 Discrete Probability Distributions 5. Which of the following about the binomial distribution is not a true statement? a) The probability of the event of interest must be constant from trial to trial. b) Each outcome is independent of the other. c) Each outcome may be classified as either "event of interest" or "not event of interest." d) The variable of interest is continuous. d TYPE: MC DIFFICULTY: Easy, properties 6. In a binomial distribution a) the variable X is continuous. b) the probability of event of interest π 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. b TYPE: MC DIFFICULTY: Easy, properties 7. Whenever π = 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. a TYPE: MC DIFFICULTY: Moderate, properties 8. Whenever π = 0.1 and n is small, the binomial distribution will be a) symmetric. b) right-skewed. c) left-skewed. d) None of the above. b TYPE: MC DIFFICULTY: Moderate, properties Copyright 2015 Pearson Education
Discrete Probability Distributions 5-3 9. If n = 10 and π = 0.70, then the mean of the binomial distribution is a) 0.07 b) 1.45. c) 7.00 d) 14.29 c TYPE: MC DIFFICULTY: Easy, mean 10. If n = 10 and π = 0.70, then the standard deviation of the binomial distribution is a) 0.07 b) 1.45 c) 7.00 d) 14.29 b TYPE: MC DIFFICULTY: Easy, standard deviation 11. If the outcomes of a 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. c TYPE: MC DIFFICULTY: Easy, 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. a TYPE: MC DIFFICULTY: Difficult, properties
5-4 Discrete Probability Distributions 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. 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. a TYPE: MC DIFFICULTY: Difficult, properties 15. What type of probability distribution will most likely be used to analyze the number of blue chocolate chips per bag in the following problem? The quality control manager of a candy plant is inspecting a batch of chocolate chip bags. When the production process is in control, the mean number of blue chocolate chips per bag is 6.0. The manager is interested in analyzing the probability that any particular bag being inspected has fewer than 5.0 blue chocolate chips. a) binomial distribution. b) Poisson distribution. c) hypergeometric distribution. d) none of the above. b TYPE: MC DIFFICULTY: Moderate, properties Copyright 2015 Pearson Education
Discrete Probability Distributions 5-5 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. c TYPE: MC DIFFICULTY: Moderate, 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. c TYPE: MC DIFFICULTY: Moderate, 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. b TYPE: MC DIFFICULTY: Moderate, properties
5-6 Discrete Probability Distributions 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. a TYPE: MC DIFFICULTY: Moderate, 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. d) does not depend on the covariance. 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. c TYPE: MC DIFFICULTY: Moderate, properties Copyright 2015 Pearson Education
Discrete Probability Distributions 5-7 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. b TYPE: MC DIFFICULTY: Moderate, 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. a TYPE: MC DIFFICULTY: Moderate, properties 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 mean cost for all weekly rat purchases is $13.00. d) The rat cost that occurs more often than any other is $13.00. c TYPE: MC DIFFICULTY: Easy KEYWORDS: mean (expected value), probability distribution
5-8 Discrete Probability Distributions 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 b TYPE: MC DIFFICULTY: Moderate KEYWORDS: mean (expected value), probability distribution 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. c TYPE: MC DIFFICULTY: Easy KEYWORDS: mean (expected value), Poisson distribution 27. True or False: The Poisson distribution can be used to model a continuous random variable. False TYPE: TF DIFFICULTY: Easy, properties 28. True or False: Another name for the mean of a probability distribution is its expected value. True TYPE: TF DIFFICULTY: Easy KEYWORDS: mean (expected value) Copyright 2015 Pearson Education
Discrete Probability Distributions 5-9 29. True or False: The number of customers arriving at a department store in a 5-minute period has a binomial distribution. False TYPE: TF DIFFICULTY: Easy, properties 30. True or False: The number of customers arriving at a department store in a 5-minute period has a Poisson distribution. True TYPE: TF DIFFICULTY: Easy, 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. False TYPE: TF DIFFICULTY: Easy, 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. True TYPE: TF DIFFICULTY: Easy, properties 33. True or False: The diameters of 10 randomly selected bolts have a binomial distribution. False TYPE: TF DIFFICULTY: Easy, properties 34. True or False: The largest value that a Poisson random variable X can have is n. False TYPE: TF DIFFICULTY: Moderate, properties
5-10 Discrete Probability Distributions 35. True or False: In a Poisson distribution, the mean and standard deviation are equal. False TYPE: TF DIFFICULTY: Moderate, properties 36. True or False: In a Poisson distribution, the mean and variance are equal. True TYPE: TF DIFFICULTY: Moderate, properties 37. True or False: If π remains constant in a binomial distribution, an increase in n will increase the variance. True TYPE: TF DIFFICULTY: Moderate, properties 38. True or False: If π remains constant in a binomial distribution, an increase in n will not change the mean. False TYPE: TF DIFFICULTY: Moderate, properties 39. True or False: Suppose that a judge s decisions follow a binomial distribution and that his verdict is incorrect 10% of the time. In his next 10 decisions, the probability that he makes fewer than 2 incorrect verdicts is 0.736. True TYPE: TF DIFFICULTY: Moderate, probability 40. True or False: Suppose that the number of airplanes arriving at an airport per minute is a Poisson process. The mean number of airplanes arriving per minute is 3. The probability that exactly 6 planes arrive in the next minute is 0.05041. True TYPE: TF DIFFICULTY: Easy, probability Copyright 2015 Pearson Education
Discrete Probability Distributions 5-11 41. True or False: The covariance between two investments is equal to the sum of the variances of the investments. 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. 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. False TYPE: TF DIFFICULTY: Moderate KEYWORDS: portfolio, mean (expected value) 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. 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. 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. True TYPE: TF DIFFICULTY: Easy KEYWORDS: portfolio, mean (expected value)
5-12 Discrete Probability Distributions SCENARIO 5-1 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 Scenario 5-1, the probability that both sound an alarm in the presence of smoke is. 0.64 48. Referring to Scenario 5-1, the probability that neither sound an alarm in the presence of smoke is. 0.04 49. Referring to Scenario 5-1, the probability that at least one sounds an alarm in the presence of smoke is. 0.96 TYPE: FI DIFFICULTY: Difficult SCENARIO 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 Scenario 5-2, the probability that all 3 businesses succeed is. 0.216 51. Referring to Scenario 5-2, the probability that all 3 businesses fail is. 0.064 Copyright 2015 Pearson Education
Discrete Probability Distributions 5-13 52. Referring to Scenario 5-2, the probability that at least 1 business succeeds is. 0.936 53. Referring to Scenario 5-2, the probability that exactly 1 business succeeds is. 0.288 54. If X has a binomial distribution with n = 4 and p = 0.3, then P(X = 1) =. 0.4116 55. If X has a binomial distribution with n = 4 and p = 0.3, then P(X > 1) =. 0.3483 56. If X has a binomial distribution with n = 5 and p = 0.1, then P(X = 2) =. 0.0729 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. 0.0768 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. 0.9898
5-14 Discrete Probability Distributions 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. 0.3456 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. 0.2592 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 2 or fewer prefer brand C is. 0.3174 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. 0.3370 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 fewer than 2 prefer brand C is. 0.0870 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 mean number that you would expect to prefer brand C is. 3, mean Copyright 2015 Pearson Education
Discrete Probability Distributions 5-15 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. 1.2, variance SCENARIO 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 network. X 0 1 2 3 P(X) 0.35 0.35 0.25 0.05 66. Referring to Scenario 5-3, the probability of no retransmissions is. 0.35 KEYWORDS: probability distribution 67. Referring to Scenario 5-3, the probability of at least one retransmission is. 0.65 KEYWORDS: probability distribution 68. Referring to Scenario 5-3, the mean or expected value for the number of retransmissions is. 1.0 KEYWORDS: probability distribution, mean (expected value), 69. Referring to Scenario 5-3, the variance for the number of retransmissions is. 0.80 KEYWORDS: probability distribution, variance 70. Referring to Scenario 5-3, the standard deviation of the number of retransmissions is. 0.894 KEYWORDS: probability distribution, standard deviation
5-16 Discrete Probability Distributions 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. 0.2373 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. 0.3955 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. 0.7627 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. 0.8965 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. 1.25, mean (expected value), Copyright 2015 Pearson Education
Discrete Probability Distributions 5-17 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. 0.9375, variance 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. 0.968, standard deviation SCENARIO 5-4 The following table contains the probability distribution for X = the number of traffic accidents reported in a day in a small city in the Midwest. X 0 1 2 3 4 5 P(X) 0.10 0.20 0.45 0.15 0.05 0.05 78. Referring to Scenario 5-4, the probability of 3 accidents is. 0.15 KEYWORDS: probability distribution 79. Referring to Scenario 5-4, the probability of at least 1 accident is. 0.90 KEYWORDS: probability distribution 80. Referring to Scenario 5-4, the mean or expected value of the number of accidents is. 2.0 KEYWORDS: probability distribution, mean,
5-18 Discrete Probability Distributions 81. Referring to Scenario 5-4, the variance of the number of accidents is. 1.4 KEYWORDS: probability distribution, variance 82. Referring to Scenario 5-4, the standard deviation of the number of accidents is. 1.18 KEYWORDS: probability distribution, standard deviation 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. 0.0892 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. 0.9380 TYPE: FI DIFFICULTY: Difficult 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. 0.9975 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. 0.0174 Copyright 2015 Pearson Education
Discrete Probability Distributions 5-19 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. 0.1487 TYPE: FI DIFFICULTY: Difficult 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. 6 89. The number of 911 calls in a small city has a Poisson distribution with a mean of 10 calls a day. The probability of seven 911 calls in a day is. 0.0901 90. The number of 911 calls in a small city has a Poisson distribution with a mean of 10 calls a day. The probability of seven or eight 911 calls in a day is. 0.2027 91. The number of 911 calls in a small city has a Poisson distribution with a mean of 10 calls a day. The probability of 2 or more 911 calls in a day is. 0.9995 92. The number of 911 calls in a small city 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. 3.16
5-20 Discrete Probability Distributions 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? 0.1213 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? 0.1379 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? 0.1538 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? 0.8462 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 no more than 1 of the members chosen for the team have some prior competition experience? 0.5934 Copyright 2015 Pearson Education
Discrete Probability Distributions 5-21 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? 0.3297 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. 0.8811 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. 0.1487 101. Current estimates suggest that 75% of the home-based computers in a foreign country have access to on-line services. Suppose 20 people with home-based computers were randomly and independently sampled. Find the probability that fewer than 10 of those sampled currently have access to on-line services. 0.0039 102. Current estimates suggest that 75% of the home-based computers in a foreign country have access to on-line services. Suppose 20 people with home-based computers were randomly and independently sampled. Find the probability that more than 9 of those sampled currently do not have access to on-line services. 0.0139
5-22 Discrete Probability Distributions 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. 0.0008 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. 0.2264 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. 0.1133 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. 0.3837 Copyright 2015 Pearson Education
Discrete Probability Distributions 5-23 107. 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. 0.1586 SCENARIO 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 Scenario 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? 0.6863 109. Referring to Scenario 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? 0.0773 110. Referring to Scenario 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? 0.0802 111. Referring to Scenario 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? 0.09388
5-24 Discrete Probability Distributions 112. Referring to Scenario 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? 0.9834 113. Referring to Scenario 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? 0.6863 114. Referring to Scenario 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? 0.001543 115. Referring to Scenario 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? 0.01661 116. Referring to Scenario 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? 0.09388 Copyright 2015 Pearson Education
Discrete Probability Distributions 5-25 SCENARIO 5-6 The quality control manager of Green Bulbs Inc. is inspecting a batch of energy saving compact fluorescent light bulbs. When the production process is in control, the mean number of bad bulbs per shift is 6.0. 117. Referring to Scenario 5-6, what is the probability that any particular shift being inspected has produced 4.0 bad bulbs. 0.1339 118. Referring to Scenario 5-6, what is the probability that any particular shift being inspected has produced fewer than 5.0 bad bulbs. 0.2851 119. Referring to Scenario 5-6, what is the probability that any particular shift being inspected has produced at least 6.0 bad bulbs. 0.5543 120. Referring to Scenario 5-6, what is the probability that any particular shift being inspected has produced between 5.0 and 8.0 inclusive bad bulbs. 0.5622 121. Referring to Scenario 5-6, what is the probability that any particular shift being inspected has less than 5.0 or more than 8.0 bad bulbs. 0.4378
5-26 Discrete Probability Distributions SCENARIO 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: Returns Probability Neighborhood A Neighborhood B.25 $22,500 $30,500.40 $10,000 $25,000.35 $40,500 $10,500 122. Referring to Scenario 5-7, what is the expected value gain for the house in neighborhood A? $ 12,550 KEYWORDS: portfolio, mean (expected value), 123. Referring to Scenario 5-7, what is the expected value gain for the house in neighborhood B? $ 21,300 KEYWORDS: portfolio, mean (expected value), 124. Referring to Scenario 5-7, what is the variance of the gain in value for the house in neighborhood A? 583,147,500 dollars 2 KEYWORDS: portfolio, variance 125. Referring to Scenario 5-7, what is the variance of the gain in value for the house in neighborhood B? 67,460,000 dollars 2 KEYWORDS: portfolio, variance 126. Referring to Scenario 5-7, what is the standard deviation of the value gain for the house in neighborhood A? $24,148.45 KEYWORDS: portfolio, standard deviation Copyright 2015 Pearson Education
Discrete Probability Distributions 5-27 127. Referring to Scenario 5-7, what is the standard deviation of the value gain for the house in neighborhood B? $8,213.40 KEYWORDS: portfolio, standard deviation 128. Referring to Scenario 5-7, what is the covariance of the two houses? 190,040,000 KEYWORDS: portfolio, covariance 129. Referring to Scenario 5-7, what is the expected value gain if you invest in both houses? $ 33,850 KEYWORDS: portfolio, mean of the sum 130. Referring to Scenario 5-7, what is the total variance of value gain if you invest in both houses? 270,527,500 KEYWORDS: portfolio, variance of the sum 131. Referring to Scenario 5-7, what is the total standard deviation of value gain if you invest in both houses? $ 16,447.72 KEYWORDS: portfolio, standard deviation of the sum 132. Referring to Scenario 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? $ 16,925 KEYWORDS: portfolio, mean (expected value)
5-28 Discrete Probability Distributions 133. Referring to Scenario 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? $ 8,223.86 KEYWORDS: portfolio, standard deviation, risk 134. Referring to Scenario 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? $ 20,425 KEYWORDS: portfolio, mean (expected value) 135. Referring to Scenario 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? $ 5,125.12 KEYWORDS: portfolio, standard deviation, risk 136. Referring to Scenario 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? $ 18,675 KEYWORDS: portfolio, mean (expected value) 137. Referring to Scenario 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? $ 2,392.04 KEYWORDS: portfolio, standard deviation, risk Copyright 2015 Pearson Education
Discrete Probability Distributions 5-29 138. Referring to Scenario 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? $ 15,175 KEYWORDS: portfolio, mean (expected value) 139. Referring to Scenario 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? $ 14,560.11 KEYWORDS: portfolio, standard deviation, risk 140. Referring to Scenario 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? $ 13,425 KEYWORDS: portfolio, mean (expected value) 141. Referring to Scenario 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? $ 20,947.96 KEYWORDS: portfolio, standard deviation, risk 142. Referring to Scenario 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? 30% KEYWORDS: portfolio, mean (expected value), standard deviation, risk, coefficient of variation
5-30 Discrete Probability Distributions 143. Referring to Scenario 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? 10% KEYWORDS: portfolio, mean (expected value) 144. Referring to Scenario 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? 30% KEYWORDS: portfolio, standard deviation, risk SCENARIO 5-8 Two different designs on a new line of winter jackets for the coming winter are available for your manufacturing plants. Your profit (in thousands of dollars) will depend on the taste of the consumers when winter arrives. The probability of the three possible different tastes of the consumers and the corresponding profits are presented in the following table. Probability Taste Design A Design B 0.2 more conservative 180 520 0.5 no change 230 310 0.3 more liberal 350 270 145. Referring to Scenario 5-8, the table above is called the for the two designs. probability distribution KEYWORDS: probability distribution 146. Referring to Scenario 5-8, what is your expected profit when Design A is chosen? $256 thousands or $256,000 KEYWORDS: portfolio, mean (expected value) Copyright 2015 Pearson Education
Discrete Probability Distributions 5-31 147. Referring to Scenario 5-8, what is your expected profit when Design B is chosen? $340 thousands or $340,000 KEYWORDS: portfolio, mean (expected value) 148. Referring to Scenario 5-8, what is the variance of your profit when Design A is chosen? 2 4,144 1, 000 or 4,144,000,000 KEYWORDS: portfolio, variance 149. Referring to Scenario 5-8, what is the variance of your profit when Design B is chosen? 2 8,400 1,000 or 8,400,000,000 KEYWORDS: portfolio, variance 150. Referring to Scenario 5-8, what is the standard deviation of your profit when Design A is chosen? $64.37391 thousands or $64,373.91 KEYWORDS: portfolio, standard deviation 151. Referring to Scenario 5-8, what is the standard deviation of your profit when Design B is chosen? $91.65151 thousands or $91,651.51 KEYWORDS: portfolio, standard deviation 152. Referring to Scenario 5-8, what is the covariance of the profits from the two different designs? 2 4,320 1, 000 or 4,320,000,000 KEYWORDS: portfolio, covariance
5-32 Discrete Probability Distributions 153. Referring to Scenario 5-8, what is the expected profit if you increase the shift of your production lines and choose to produce both designs? $596 thousands or $596,000 KEYWORDS: portfolio, mean of the sum 154. Referring to Scenario 5-8, what is the total variance of the profit if you increase the shift of your production lines and choose to produce both designs? 2 3,904 1, 000 or 3,904,000,000 KEYWORDS: portfolio, variance of the sum 155. Referring to Scenario 5-8, what is the total standard deviation of the profit if you increase the shift of your production lines and choose to produce both designs? $62.48200 thousands or $62,482.00 KEYWORDS: portfolio, standard deviation of the sum 156. Referring to Scenario 5-8, if you decide to choose Design A for half of the production lines and Design B for the other half, what is your expected profit? $ 298 thousands or $298,000 KEYWORDS: portfolio, mean (expected value) 157. Referring to Scenario 5-8, if you decide to choose Design A for half of the production lines and Design B for the other half, what is the risk of your investment? $31.241 thousands or $31,241 KEYWORDS: portfolio, standard deviation, risk 158. Referring to Scenario 5-8, if you decide to choose Design A for half of the production lines and Design B for the other half, what is the coefficient of variation of your investment? 10.48% KEYWORDS: portfolio, coefficient of variation Copyright 2015 Pearson Education
Discrete Probability Distributions 5-33 159. Referring to Scenario 5-8, if you decide to choose Design A for 10% of the production lines and Design B for the remaining production lines, what is the expected profit? $ 331.6 thousands or $331,600 KEYWORDS: portfolio, mean (expected value) 160. Referring to Scenario 5-8, if you decide to choose Design A for 10% of the production lines and Design B for the remaining production lines, what is the risk of your investment? $77.89634 thousands or $77,896.34 KEYWORDS: portfolio, standard deviation, risk 161. Referring to Scenario 5-8, if you decide to choose Design A for 10% of the production lines and Design B for the remaining production lines, what is the coefficient of variation of your investment? 23.49% KEYWORDS: portfolio, coefficient of variation 162. Referring to Scenario 5-8, if you decide to choose Design A for 30% of the production lines and Design B for the remaining production lines, what is the expected profit? $314.8 thousands or $314,800 KEYWORDS: portfolio, mean (expected value) 163. Referring to Scenario 5-8, if you decide to choose Design A for 30% of the production lines and Design B for the remaining production lines, what is the risk of your investment? $51.71615 thousands or $51,716.15 KEYWORDS: portfolio, standard deviation, risk 164. Referring to Scenario 5-8, if you decide to choose Design A for 30% of the production lines and Design B for the remaining production lines, what is the coefficient of variation of your investment? 16.42% KEYWORDS: portfolio, coefficient of variation
5-34 Discrete Probability Distributions 165. Referring to Scenario 5-8, if you decide to choose Design A for 70% of the production lines and Design B for the remaining production lines, what is the expected profit? $281.2 thousands or $281,200 KEYWORDS: portfolio, mean (expected value) 166. Referring to Scenario 5-8 if you decide to choose Design A for 70% of the production lines and Design B for the remaining production lines, what is the risk of your investment? $31.17948 thousands or $31,179.48 KEYWORDS: portfolio, standard deviation, risk 167. Referring to Scenario 5-8, if you decide to choose Design A for 70% of the production lines and Design B for the remaining production lines, what is the coefficient of variation of your investment? 11.09% KEYWORDS: portfolio, coefficient of variation 168. Referring to Scenario 5-8, if you decide to choose Design A for 90% of the production lines and Design B for the remaining production lines, what is the expected profit? $264.4 thousands or $264,400 KEYWORDS: portfolio, mean (expected value) 169. Referring to Scenario 5-8 if you decide to choose Design A for 90% of the production lines and Design B for the remaining production lines, what is the risk of your investment? $51.60465 thousands or $ 51,604.65 KEYWORDS: portfolio, standard deviation, risk 170. Referring to Scenario 5-8, if you decide to choose Design A for 90% of the production lines and Design B for the remaining production lines, what is the coefficient of variation of your investment? 19.52% KEYWORDS: portfolio, coefficient of variation Copyright 2015 Pearson Education
Discrete Probability Distributions 5-35 171. Referring to Scenario 5-8, if your investment preference is to maximize your expected profit while exposing yourself to the minimal amount of risk, will you choose a production mix that will consist of 10%, 30%, 50%, 70%, or 90% of your production lines for Design A and the remaining for Design B? 50% KEYWORDS: portfolio, mean (expected value), standard deviation, risk, coefficient of variation NOTE: 50% yields the lowest coefficient of variation 172. Referring to Scenario 5-8, if your investment preference is to maximize your expected profit and not worry at all about the risk that you have to take, will you choose a production mix that will consist of 10%, 30%, 50%, 70%, or 90% of your production lines for Design A and the remaining for Design B? 10% KEYWORDS: portfolio, mean (expected value) 173. Referring to Scenario 5-8, 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 profit, will you choose a production mix that will consist of 10%, 30%, 50%, 70%, or 90% of your production lines for Design A and the remaining for Design B? 70% KEYWORDS: portfolio, standard deviation, risk SCENARIO 5-9 A major hotel chain keeps a record of the number of mishandled bags per 1,000 customers. In a recent year, the hotel chain had 4.06 mishandled bags per 1,000 customers. Assume that the number of mishandled bags has a Poisson distribution. 174. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have no mishandled bags? 0.0172
5-36 Discrete Probability Distributions 175. Referring to Scenario 5-9, what is the probability that in the next 1,000 custoers, the chain will have at least one mishandled bags? 0.9828 176. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have at least two mishandled bags? 0.9127 177. Referring to Scenario 5-9, what is the probability that in the next 1,000 customer, the hotel chain will have no more than three mishandled bags? 0.4218 178. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have no more than four mishandled bags? 0.6171 179. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have fewer than six mishandled bags? 0.7757 180. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have fewer than eight mishandled bags? 0.9452 Copyright 2015 Pearson Education
Discrete Probability Distributions 5-37 181. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have more than eight mishandled bags? 0.02320 182. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have more than ten mishandled bags? 0.003172 183. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have between two and four inclusive mishandled bags? 0.5298 184. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have more than five but less than eight mishandled bags? 0.1695 185. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have less than two or more than eight mishandled bags? 0.1105 186. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have no more than two or at least eight mishandled bags? 0.2842
5-38 Discrete Probability Distributions 187. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have less than two and more than eight mishandled bags? 0 188. Referring to Scenario 5-9, what is the probability that in the next 1,000 customers, the hotel chain will have no more than two and at least eight mishandled bags? 0 SCENARIO 5-10 An accounting firm in a college town usually recruits employees from two of the universities in town. This year, there are fifteen graduates from University A and five from University B and the firm decides to hire six new employees from the two universities. 189. Referring to Scenario 5-10, what is the probability that two of the new employees will be from University A? 0.0135 or 1.35% 190. Referring to Scenario 5-10, what is the probability that all six of the new employees will be from University A? 0.1291 or 12.91% 191. Referring to Scenario 5-10, what is the probability that none of the new employees will be from University B? 0.1291 or 12.91% Copyright 2015 Pearson Education