CHAPTER 4 Probability Multiple-Choice Questions QUESTIONS 1 THROUGH 11 ARE BASED ON THE FOLLOWING INFORMATION: Suppose you roll a pair of dice. Let A be the event that you observe an even number. Let B be the event that you observe a number greater than seven. 1. What is the intersection of events A and B? A) [8, 10, 12] B) [7, 8, 9, 10, 11, 12] C) [2, 4, 6, 7, 8, 9, 10, 11, 12] D) [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] ANSWER: A 2. What is the union of events A and B? A) [8, 9, 10, 11, 12] B) [8, 10, 12] C) [2, 4, 6, 8, 9, 10, 11, 12] D) [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] ANSWER: C 3. Which of the following statements is true? A) The events A and B are mutually exclusive. B) The intersection of A and B is the set [6, 8, 10, 12]. C) The events A and B are collectively exhaustive. 85
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CHAPTER 4
Probability
Multiple-Choice Questions
QUESTIONS 1 THROUGH 11 ARE BASED ON THE FOLLOWING INFORMATION:Suppose you roll a pair of dice. Let A be the event that you observe an even number. Let B be the event that you observe a number greater than seven.
A) The events A and B are mutually exclusive.B) The intersection of A and B is the set [6, 8, 10, 12].C) The events A and B are collectively exhaustive.D) None of the above.ANSWER: D
12. Suppose you roll a pair of dice. Let A be the event that you roll an even number. Let B be the event that you roll an odd number. Which of the following statements is true?
A) The events A and B are not mutually exclusive.B) The intersection of A and B is the empty set .C) The events A and B are not collectively exhaustive.D) The complement of event B is the set [1, 3, 5, 7, 9, 11].ANSWER: B
13. Which of the following statements is always true for any two events A and B defined on a sample space S?
A) The complement of event A is event B.B) The intersection of A and B is the set of all basic outcomes in either A or B.C) If events A and B are mutually exclusive, then .
D) If events A and B are collectively exhaustive, then .ANSWER: D
14. Which of the following statements is always true for any two events A and B defined on a sample space S?
A) If the complement of event A is the empty set, then event A is the sample space S.B) If the union of events A and B is not the empty set, then .
C) If events A and B are mutually exclusive, then .
D) If events A and B are collectively exhaustive, then .ANSWER: A
15. Which of the following statements is true for any two events A and B defined on a sample space S?
A) If the intersection of events A and B is the empty set, then A and B are collectively exhaustive.
B) If the union of events A and B is the empty set, then each of A and B is the empty set.
C) If events A and B are collectively exhaustive, then .D) If events A and B are mutually exclusive and collectively exhaustive, then the union
of A and B is not necessarily the sample space.ANSWER: B
16. Which of the following is the appropriate definition for the union of two events A and B?
A) The set of all basic outcomes contained within both A and B.B) The set of all basic outcomes in either A or B, or both.C) The set of all possible outcomes.D) None of the above.
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ANSWER: B
17. Which of the following is the appropriate definition for the intersection of two events A and B defined on a sample space?
A) The set of all basic outcomes contained within both A and B.B) The set of all basic outcomes in either A or B, or both.C) The set of all possible outcomes.D) None of the above.ANSWER: A
18. Two events A and B defined on a sample space S are said to be collectively exhaustive if
A) .
B) .
C) .
D) .ANSWER: D
19. Two events A and B defined on a sample space S are said to be mutually exclusive if
A) .
B) .
C) .
D) .ANSWER: A
QUESTIONS 20 THROUGH 22 ARE BASED ON THE FOLLOWING INFORMATION:The production manager at a local manufacturing plant is worried about work stoppages in his four production lines. In particular, the manager is evaluating the likelihood of the next stoppage occurring on production line 1.
20. The manager figures that each of the lines is equally likely to have the next stoppage; therefore, the probability that production line 1 is stopped next is 25%. This is an example of:
A) Subjective probability.B) Classical probability.C) Relative frequency probability.D) Bayesian probability.ANSWER: B
21. Based on previous stoppage records, the manager figures that the probability that production line 1 is stopped next is 25%. This is an example of:
A) Subjective probability.B) Classical probability.C) Relative frequency probability.D) Bayesian probability.ANSWER: C
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22. Based on his knowledge of the causes of stoppages, the manager states that the probability that production line 1 is stopped next is 25%. This is an example of:
A) Subjective probability.B) Classical probability.C) Relative frequency probability.D) Bayesian probability.ANSWER: A
QUESTIONS 23 THROUGH 25 ARE BASED ON THE FOLLOWING INFORMATION:The finishing process on new furniture leaves slight blemishes. The table shown below displays a manager’s probability assessment of the number of blemishes in the finish of new furniture.
Number of Blemishes 0 1 2 3 4 5Probability 0.34 0.25 0.19 0.11 0.07 0.04
23. Let event A be that there are more than three blemishes and let event B be that there are four or fewer blemishes. Which of the following statements is true?
A) P( ) = 0.18.
B) P( ) = 0.07.C) Events A and B are collectively exhaustive.D) Events A and B are mutually exclusive.ANSWER: C
24. Let event A be that there are more than two blemishes and let event B be that there are four or fewer blemishes. Which of the following statements is true?
A) P( ) = 0.18.
B) P( ) = 0.07.
C) P( ) = 0.58.
D) P( ) = 0.89.ANSWER: A
25. Let A be the event that there is at least one blemish and let event B be that there at most three blemishes. Which of the following statements is true?
A) P( ) = 0.76.
B) P( ) = 0.55.
C) P( ) = 0.30.
D) P( ) = 0.96.ANSWER: B
QUESTIONS 26 AND 27 ARE BASED ON THE FOLLOWING INFORMATION:Ted’s Surfboard Shop makes surfboards by hand. The number of surfboards that Ted makes during a week depends on the wave conditions. Ted has estimated the following probabilities for surfboard production for the next week.
Number of Surfboards 5 6 7 8 9 10
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Probability 0.13 0.22 0.31 0.17 0.13 0.04
26. Let event A be that Ted produces more than seven surfboards and let event B be that he produces exactly six surfboards. Which of the following statements is true?
A) P( )=0.31B) Events A and B are collectively exhaustive.C) P( )=0.44D) Events A and B are mutually exclusiveANSWER: D
27. Let event A be that Ted produces more than six surfboards, and let event B be that he produces less than eight surfboards. Which of the following statements is true?
A) P( ) = 0.75B) Events A and B are collectively exhaustive.C) P( ) = 0.53D) Events A and B are mutually exclusive.ANSWER: B
QUESTIONS 28 AND 29 ARE BASED ON THE FOLLOWING INFORMATION:A multiple choice quiz has five questions, each with five answers, A through E. Assume you just guess on all of the questions.
28. What is the probability that you guessed on all five questions right?
A) 0.00032B) 0.03125C) 0.20D) 0.50ANSWER: A
29. What is the probability that you get exactly three questions right?
A) 0.00032B) 0.0016 C) 0.008D) 0.04ANSWER: C
30. The probability that an employee at a company uses illegal drugs is 0.08. The probability than an employee is male is 0.55. Assuming that these events are independent, what is the probability that a randomly chosen employee is a male drug user?
A) 0.742B) 0.145C) 0.044 D) 0.006ANSWER: C
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31. Your have inherited a racing horse. The trainer of the horse tells you that the odds in favor of you winning the Kentucky Derby are 1 to 9. That means that the probability of the horse actually winning is:
A) 0.11.B) 0.10.C) 0.09. D) 0.08.ANSWER: B
QUESTIONS 32 THROUGH 34 ARE BASED ON THE FOLLOWING INFORMATION:A recent marketing survey tried to relate a consumer’s awareness of a new marketing campaign with their rating of the product. Consumers rated their awareness as low, medium, or high, and rated the product as poor, fair, or good. The results are presented below.
AwarenessLow Medium High
Poor 0.10 0.15 0.07Rating Fair 0.06 0.11 0.06
Good
0.07 0.11 0.27
32. What is the probability that a consumer had low awareness?
A) 0.10B) 0.14C) 0.23D) 0.071ANSWER: C
33. What is the probability that a consumer who ranked the product as fair had a high awareness of the add campaign?
A) 0.06B) 0.26C) 0.23D) 0.40ANSWER: B
34. What is the probability that a consumer who had high awareness of the ad campaign ranked the product as good?
A) 0.675B) 0.385C) 0.775D) 0.325ANSWER: A
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35. A junior executive looking at his business attire in his closet notes that he has five suits, six shirts and three pairs of shoes. He is going on a business trip and needs to take two of each. How many different combinations of outfits could he take?
A) 680B) 320C) 450D) 224ANSWER: C
36. It was found that 84% of all stockbrokers drink more than three 8 oz. cups of coffee each day. Furthermore, 64% eat at least one candy bar each day. Finally, half of those stockbrokers who do not drink coffee, eat a candy bar. What is the probability that a stockbroker who eats candy also drinks coffee?
A) 0.360B) 0.560C) 0.875D) 0.280ANSWER: C
37. A junior executive looking at his business attire in his closet notes that he has eight suits, six shirts and four pairs of shoes. He is going on a business trip and needs to take two of each. How many different combinations of outfits could he take?
A) 2240B) 3320C) 1680D) 2520ANSWER: D
38. The purchasing agent for a municipality has contracted with a local car dealer to purchase four cars. The dealer has 25 cars on his lot; 10 red, 7 blue, 6 white and 2 purple. If the purchasing agent has no control over the colors he receives, what is the probability that he receives at least one of the purple cars?
A) 0.33B) 0.30C) 0.36D) 0.39ANSWER: B
39. As the office manager for a medical office with ten doctors, you are responsible for developing the roster for the on-call shift for the next three nights. How many different ways could you assign different doctors to the next three nights?
A) 1020B) 840C) 720D) 640ANSWER: C
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40. Which of the following statements is true?
A) If events A and B are complements, then the intersection of A and the complement of B is the sample space.
B) If events A and B are mutually exclusive, then the intersection of A and B is the empty set.
C) If events A and B are mutually exclusive, then the union of A and B is the sample space.
D) If events A and B are mutually exclusive, then the union of A and B is the empty set.ANSWER: B
41. A recent survey found that 14% of secretaries have experienced some form of wrist pain from typing. 6% of all secretaries have both experienced some form of wrist pain from typing and taken aspirin on a daily basis. What is the probability that a secretary who has wrist pain takes aspirin on a daily basis?
A) 0.429B) 0.915C) 0.571D) 0.085ANSWER: A
42. In a recent article it was reported that 35.4% of all high school students smoke cigarettes. 65% of these students plan on going to college. What is the probability that a randomly selected student smokes cigarettes and plans on going to college?
A) 0.354B) 0.230C) 0.124D) 0.412ANSWER: B
43. An office of six people is plagued by high absenteeism. It is thought that the probability that an employee is absent on a particular day is 0.03. Assuming that the event that one person is absent on a particular day is independent of the absence of any other employee, what is the probability that at least one employee is absent tomorrow?
A) 0.121B) 0.180C) 0.150D) 0.167ANSWER: D
44. A gumball machine has three different colored gumballs: red, blue and yellow. If you buy 3 gumballs, how many different combinations of colors could you buy?
A) 9B) 15C) 10D) 6ANSWER: C
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45. A gumball machine has six red gumballs: four blue gumballs and three yellow gumballs. If you buy three gumballs, what is the probability that you get three different colors?
A) 0.252B) 0.167C) 0.125D) 0.244ANSWER: A
QUESTIONS 46 THROUGH 49 ARE BASED ON THE FOLLOWING INFORMATION:A survey of recent e-commerce start-up firms was undertaken at an industry convention. Representatives of the firm where asked for the geographic location of the firm as well as the firm’s outlook for growth in the coming year. The results are provided below.
46. What is the probability that one of these start-up firms was from the Northeast?
A) 0.04B) 0.12C) 0.49D) 0.33ANSWER: B
47. Are the events “firm from the South” and “expects high growth” statistically independent?
A) YesB) NoC) Unable to tell from the dataD) MaybeANSWER: A
48. If the firm interviewed was from the West, what is the probability that it expected medium or high growth?
A) 0.24B) 0.35C) 0.16D) 0.46ANSWER: D
49. If the firm interviewed was expecting medium or high growth, what is the probability of the firm being located in the West?
QUESTIONS 50 THROUGH 53 ARE BASED ON THE FOLLOWING INFORMATION:In a recent survey of consumer confidence, 160 respondents were classified by their level of education. The results of the survey are presented below.
50. What is the proportion of respondents who had medium or high confidence.
A) 0.330B) 0.271C) 0.719D) 0.670ANSWER: C
51. What proportion of respondents had at least some college education and had high confidence?
A) 0.131B) 0.242C) 0.558D) 0.175ANSWER: A
52. What proportion of respondents who had at least some college education also had high confidence?
A) 0.239B) 0.210C) 0.364D) 0.636ANSWER: A
53. Are the events “had a college education” and “had high confidence” statistically independent?
A) Yes.B) No.C) Maybe.D) There is not sufficient information to determine.ANSWER: B
54. Consider two events A and B. Which of the following statements is true?
A) If the probability of A given B is 0.4, then the probability of A given the complement of B is 0.6.
B) If the probability of A given B is 0.4, then the probability of the complement of A given the complement of B is 0.6.
C) If the probability of A given B is 0.4, then the probability of the complement of A given B is 0.6
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EducationHigh School Some College College
Low 13 17 15Confidence Medium 27 22 13
High 32 14 7
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D) If the probability of A given B is 0.4 and the probability of A is 0.4, then events A and B are mutually exclusive.
ANSWER: C
55. If the odds for you getting a credit card solicitation in the mail this month are 1 to 4, then which of the following statements is true?
A) The probability of getting a credit card solicitation is 0.25.B) The probability of getting a credit card solicitation is 0.20.C) The probability of getting a credit card solicitation is 0.80.D) The probability of getting a credit card solicitation is 0.75.ANSWER: B
56 Consider two events A and B. Which of the following statements is true?
A) If the probability of A given B is 0.4 and the probability of B is 0.6, then the probability of A is 0.2.
B) If the probability of A given B is 0.4 and the probability of A is 0.4, then events A and B are statistically independent.
C) If the probability of A given B is 0.4 and the probability of the union of A and B is 0.7, then the probability of A is 0.3.
D) If the probability of A given B is 0.4 and the probability of B is 0.6, then the probability of A is 0.2.
ANSWER: B
57. In a survey of 100 large corporations, it was found that 72% offer some form of tuition assistance plan for their workers. 64% of the 100 corporations offer both a tuition assistance plan as well as provide dental insurance for dependents. What is the probability that a corporation that offers a tuition assistance plan also offers dental insurance for dependents?
A) 0.89B) 0.46C) 0.64D) 0.68ANSWER: A
QUESTIONS 58 THROUGH 60 ARE BASED ON THE FOLLOWING INFORMATION:In a recent survey of college students, students where asked about their use of the Internet for research. The information was displayed in the table below. Let A be the event that the student was a business major. Let B be the event that the student uses the Internet for research.
58. Which of the following is true? A) P(A | B) < P(B | A).B) P(A | ) > P(B | ).
C) P(A B) = 0.85.D) Events A and B are statistically independent.ANSWER: A
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Use Internet for research Don’t Use InternetBusiness Students 0.38 0.22Education Students 0.25 0.15
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59. Which of the following is true?
A) P(A | B) = P(B | A)B) P (A | ) < P(B | )
C) P(A B) = 0.60D) Events A and B are statistically independent.ANSWER:
60. Which of the following is true? A) P(A | B) = 0.81B) P(A ) = 0.16C) Events A and B are statistically independent.D) P( | ) = 0.375ANSWER: D
61. Which of the following statements is not true?
A) Insurance companies employ the subjective approach to probability in many ways.B) In general, .
C) If is the complement of event A, then P(A) + P( ) = 1.0.D) None of the above.ANSWER: A
62. Which of the following statements are not true?
A) An event A and its complement are always mutually exclusiveB) A set of events is said to be exhaustive if it includes all the possible outcomes of an
experiment.C) The probability of event A and event B occurring must be equal to 1.D) When the events within a set are both mutually exclusive and exhaustive, the sum of
their probabilities is 1.0.ANSWER: C
63. An example of the classical approach to probability would be:
A) the estimate of number of defective parts based on previous production data.B) your estimate of the probability of a pop quiz in class on a given day.C) the annual estimate of the number of deaths of persons age 45.D) the outcomes in a card game.ANSWER: D
64. If A and B are independent events with P(A) = 0.60 and P(A | B) = 0.60, then P(B) is:
A) 1.20. B) 0.60.C) 0.36.D) Cannot be determined with the information given.ANSWER: D
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65. If a coin is tossed three times and a statistician predicts that the probability of obtaining three heads in a row is 0.125, which of the following assumptions is irrelevant to his prediction?
A) The events are dependent.B) The events are independent.C) The coin is unbiased.D) All of the above.ANSWER: A
66. Which of the following is not an example of the relative frequency approach to probability?
A) The outcome of a game of craps based on your personal experience.B) The outcome of a game of roulette based on historical data.C) The outcome of a poker game based on the draw of one card.D) The actuarial schedule for a life insurance company.ANSWER: C
67. Which of the following statements is not true?
A) Two events A and B are mutually exclusive if event A occurs and event B cannot occur.
B) If events A and B occur at the same time, then A and B intersect.C) If event A does not occur, then its complement will also not occur.D) A union of events occurs when at least one event in a group occurs.ANSWER: C
68. If P(A) = 0.20, P(B) = 0.30 and P(A B) = 0.06, then A and B are
A) dependent events.B) independent events.C) mutually exclusive events.D) complementary events.ANSWER: B
69. Two events A and B are said to be mutually exclusive if:
A) P(A | B) = 1.B) P(B | A) =1.C) P(A B) =1.
D) P(A B) = 0.ANSWER: D
70. If P(A) = 0.84, P(B) = 0.76 and P(A B) = 0.90, then P(A B) is:
A) 0.06.B) 0.14.C) 0.70.D) 0.83.ANSWER: C
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71. If P(A) = 0.25 and P(B) = 0.60, then P(A B) is:
A) 0.15.B) 0.35.C) 0.25.D) Cannot be determined from the information given.ANSWER: D
72. Which of the following statements is not true?
A) Events are independent when the occurrence of one event has no effect on the probability that another will occur.
B) When events A and B are independent, then P(A B) =P(A) + P(B).
C) When events A and B are independent, then P(A B) = P(A) P(B).
D) When events A and B are independent, then P(A B) = P(A) + P(B) – P(A) P(B).ANSWER: B
73. If the events A and B are independent with P(A) = 0.30 and P(B) = 0.40, then the probability that both events will occur simultaneously is:
A) 0.10.B) 0.12.C) 0.70.D) 0.75.ANSWER: B
74. Which of the following statements is true given that the events A and B have nonzero probabilities?
A) A and B cannot be both independent and mutually exclusiveB) A and B can be both independent and mutually exclusiveC) A and B are always independentD) A and B are always mutually exclusiveANSWER: A
75. If A and B are independent events with P(A) = 0.60 and P(B) = 0.70, then the probability that A occurs or B occurs or both occur is:
A) 1.30.B) 0.88.C) 0.42.D) 0.10.ANSWER: B
76. If A and B are independent events with P(A) = 0.20 and P(B) = 0.60, then P(A | B) is:
A) 0.20.B) 0.60.C) 0.40.D) 0.80.ANSWER: A
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QUESTIONS 77 THROUGH 82 ARE BASED ON THE FOLLOWING INFORMATION:The joint probabilities shown in a table with two rows, and and two columns, and ,
are as follows: P( ) = 0.10, P( ) = 0.30, P( ) = 0.05, and P( ) = 0.55.
77. What is P( )?
A) 0.40B) 0.60C) 0.15D) 0.85ANSWER: C
78. What is P( )?
A) 0.40B) 0.60C) 0.15D) 0.85ANSWER: D
79. What is P( | ), calculated up to two decimals ?
A) 0.33B) 0.35C) 0.65D) 0.67ANSWER: D
80. What is P( | ), calculated up to two decimals?
A) 0.33B) 0.35C) 0.65D) 0.67ANSWER: B
81. What is P( | ), calculated up to two decimals?
A) 0.33B) 0.35C) 0.65D) 0.67ANSWER: A
82. What is P( | ), calculated up to two decimals?
A) 0.33B) 0.35C) 0.65
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D) 0.67ANSWER: C
83. Bayes’ Theorem is used to compute the
A) prior probabilities.
B) probabilities of the intersection of two events.
C) probabilities of the union of two events.
D) posterior probabilities.ANSWER: D
84. Initial estimates of the probabilities of events are known as
A) joint probabilities.
B) posterior probabilities.
C) prior probabilities.
D) conditional probabilities.ANSWER: C
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True-False Questions
85. Events A and B are said to be mutually exclusive if their intersection is the empty set. ANSWER: T
86. Events A and B are said to be collectively exhaustive if their intersection is the empty set. ANSWER: F
87. Events A and B are said to be statistically independent if their intersection is the empty set. ANSWER: F
88. Events A and B are said to be statistically independent if their union is the sample space. ANSWER: F
89. The intersection of events A and B is given by all basic outcomes common to both A and B. ANSWER: T
90. The union of events A and B is given by all basic outcomes common to both A and B. ANSWER: F
91. The classical definition of probability is the limit of the proportion of times that event A occurs in a large number of trials, n. ANSWER: F
92. If the odds in favor of an event happening are 1 to 2, then the probability that event A occurs is 0.5. ANSWER: F
93. If events A and B are statistically independent, then the probability of their intersection is equal to the probability of event A times the probability of event B. ANSWER: T
94. If events A and B are mutually exclusive and collectively exhaustive, then the complement of event A is identical to event B. ANSWER: T
95. If the union of events A and B is not the empty set, then the two events are not mutually exclusive. ANSWER: F
96. The complement of the union of two events is the intersection of their complements.ANSWER: T
97. The sum of the probabilities of collectively exhaustive events must equal 1.ANSWER: F
98. The number of combinations of x objects chosen from n is equal to the number of combinations of (n-x) objects chosen from n, where (n-1).ANSWER: T
99. If A and B are two events, the probability of A, given B, is the same as the probability of, given A, if the probability of A is the same as the probability of B.
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ANSWER: T
100. If an event and its complement are equally likely to occur, the probability of that event must be 0.5.ANSWER: T
101. If events A and B are independent, then and must be independent.ANSWER: T
102. If events A and B are mutually exclusive, then and must be mutually exclusive.ANSWER: F
103. The probability of the union of two events cannot be less than the probability of their intersection.ANSWER: T
104. The conditional probability of A, given B, must be at least as large as the probability of A.ANSWER: F
105. The probability of the intersection of two events cannot be greater than either of their individual probabilities.ANSWER: T
106. An event and its complement are mutually exclusive.ANSWER: T
107. The individual probabilities of a pair of events cannot sum to more than 1.ANSWER: F
108. If two events are mutually exclusive, they must also be collectively exclusive.ANSWER: F
109. The probability of the union of two events cannot be more than the sum of their individual probabilities.ANSWER: T
110. If two events are collectively exhaustive, they must also be mutually exclusive.ANSWER: F
111. An event must be independent of its complement.ANSWER: F
112. The probability of A, given B, must be at least as large as the probability of the intersection of A and B.ANSWER: T
113. The probability of the intersection of two events cannot exceed the product of their individual probabilities.ANSWER: F
114. The posterior probability of any event must be at least as large as its prior probability.ANSWER: F
115. If either event A or event B must occur, they are called collectively exhaustive.ANSWER: T
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116. If either event A or event B must occur, they are called mutually exclusive.ANSWER: F
117. If P(A) = 0.25 and P(B) = 0.75, then A and B must be collectively exhaustive.ANSWER: F
118. If P(A B) = 1, then A and B must be collectively exhaustive.ANSWER: T
119. If P(A) = 0.35 and P(B) = 0.65, then A and B must be mutually exclusive.ANSWER: F
120. Marginal probability is the probability that a given event will occur, with no other events taken into consideration. ANSWER: T
121. Conditional probability is the probability that an event will occur, given that another event will also occur.ANSWER: F
122. If P(A B) = 0, then A and B must be collectively exhaustive.ANSWER: F
123. Bayes’ Theorem allows us to compute conditional probabilities from other forms of probability.ANSWER: T
124. In applying Bayes’ Theorem, as the prior probabilities increase, the posterior probabilities decrease.ANSWER: F
125. Bayes’ Theorem is a formula for revising an initial subjective (prior) probability value on the basis of results obtained by an empirical investigation and for, thus, obtaining a new (posterior) probability value.ANSWER: T
126. Prior probability of an event is the probability of the event before any information affecting it is given.ANSWER: T
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Basic and Applied Questions
127. The probability that an employee at a company eats lunch at the company cafeteria is 0.32. The probability that an employee is female is 0.62. The probability than an employee eats lunch at the employee cafeteria and is female is 0.21. What is the probability that a randomly chosen employee either eats at the cafeteria or is female?
ANSWER:Event A: eats in cafeteria, Event B: female.P(A) = 0.32, P(A B) = 0.21 P(B | A) = P(A B) / P(A) = 0.21 / 0.32 = 0.656
QUESTIONS 128 THROUGH 136 ARE BASED ON THE FOLLOWING INFORMATION:In a recent survey about US policy in Iraq, 62 % of the respondents said that they support US policy in Iraq. Females comprised 53% of the sample, and of the females, 46% supported US policy in Iraq. A person is selected at random.
128. What is the probability that the person we select is female and support US policy in Iraq?
QUESTIONS 137 THROUGH 140 ARE BASED ON THE FOLLOWING INFORMATION:Fred’s Surfboard Shop makes surfboards by hand. The number of surfboards that Fred makes during a week depends on the wave conditions. Fred has estimated the following probabilities for surfboard production for the next week.
Number of Surfboards 5 6 7 8 9 10Probability 0.13 0.22 0.31 0.17 0.13 0.04
Let A be the event that Fred produces more than seven surfboards. Let B be the event that Fred produces exactly six surfboards.
137. What is the probability of event A?
ANSWER:P(A) = 0.17 + 0.13 + 0.04 = 0.34
138. Are events A and B collectively exhaustive? Why?
ANSWER:No, since P(A B) S.
139. What is the probability of the complement of A?
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ANSWER:P( ) =1 – P(A) = 1 – 0.34 = 0.66
140. What is the probability of the intersection of events A and B? Why?
ANSWER:P(A B) = 0, since A and B are mutually exclusive events.
QUESTIONS 141 AND 142 ARE BASED ON THE FOLLOWING INFORMATION:You stop in at a local Internet café. The manager tells you that there are four computers in the cafe, and that the probability that one of them being available at any one time is 0.4. Assume that this probability is the same for each computer and the probability of one computer being occupied is statistically independent of any of the other computers.
141. What is the probability that all four computers are occupied?
ANSWER:
P(All computers are occupied) = = 0.1296
142. What is the probability that at least one computer is available?
ANSWER:P(at least one computer is available ) =1 – P(all computers are occupied) = 1 – 0.1296 = 0.8704
QUESTIONS 143 THROUGH 150 ARE BASED ON THE FOLLOWING INFORMATION:A recent survey examined the working arrangements of married households. It was found that 88% of the households had at least one working member. In 20% of the households with the woman not working, the man also does not work. In 40% of households in which the man doesn’t work, the woman also does not work
143. What is the probability that both man and woman are not working?
ANSWER:P(at least one working household) = 0.88 = 1 – P(both man and woman are not working)Hence, P(both man and woman are not working) = 1 - .88 = 0.12
144. What is the probability that the woman doesn’t work?
ANSWER:P(man doesn’t work | woman doesn’t work) = 0.20P(woman doesn’t work) = P(both man and woman are not working) / 0.20
= 0.12 / 0.20 = 0.60
145. What is the probability that the woman does work?
148. What is the probability that the man doesn’t work and the woman works?
ANSWER:P(man does not work) = P(man doesn’t work and woman doesn’t work)+
P(man doesn’t work and woman works)Then, 0.30 = 0.12 + P(man doesn’t work and woman works) Hence, P(man doesn’t work and woman works) = 0.30 – 0.12 = 0.18
149. What is the probability that both man and woman are working?
ANSWER:P(woman does work) = P(woman works and man works) +
P(woman works and man does not work)Then, 0.40 = P(woman works and man works) + 0.18Hence, P(woman works and man works) = 0.40 – 0.18 = 0.22
150. What is the probability that the man does work and woman does not work?
ANSWER:P(man works) = P(man works and woman works) +
P(man works and woman does nor work)Then, 0.70 = 0.22 + P(man works and woman does not work) Hence, P(man works and woman does not work) = 0.70 – 0.22 = 0.48
151. There are five men and four women working on a project. To handle one particular aspect of the project, a sub-committee needs to be formed. In the interest of balance, it is decided that the sub-committee will consist of two men and two women. How many combinations of this sub-committee are possible?
ANSWER:
# of possible combinations of this sub-committee = (10)(6) = 60
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QUESTIONS 152 THROUGH 155 ARE BASED ON THE FOLLOWING INFORMATION:A local real estate broker has collected data on the sales of 128 homes. She is interested in looking at the selling price of the house as well as the length of time the house was for sale. She prepared the following table.
152. What proportion of all houses took 30 days or more to sell?
QUESTIONS 156 THROUGH 159 ARE BASED ON THE FOLLOWING INFORMATION:A student has access to professor evaluations. Overall, he has enjoyed 70% of all classes he has taken. He finds that of the courses he has enjoyed, 13% were taught by professors with poor evaluations. 84% of the courses he has taken were taught by professors with good evaluations.
156. What is the probability that the class was taught by a professor with poor evaluation and that the student enjoyed the class?
ANSWER:Event A: Student enjoyed class Event B: Class taught by professors with good evaluationSince P(A) = 0.70 and P( | A) = 0.13
Then, P( A) = P( |A) P(A) = (0.13)(0.70) = 0.091
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Selling Price< $150K $150K but < $300K > $300K
< 30 Days 13 15 7Time on Market 30 but < 90 Days 6 11 15
> 90 Days 3 21 37
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157. What is the probability that the class was taught by a professor with good evaluation and that the student enjoyed the class?
158. What is the probability that the student enjoyed the class given that it was taught by a professor with good evaluation?
ANSWER:P(B) = 0.84P(A | B) = P(A B) / P(B) = 0.609/0.84 = 0.725
159. Suppose a student signed up for three courses next semester, all of which are taught by professors with good evaluation. What is the probability he enjoys all three?
ANSWER:
160. In a recent article it was reported that 27.3% of all college students party during weekdays, and 67% of these students plan on going to graduate school. What is the probability that a randomly-selected student party during weekdays and plans on going to graduate school?
ANSWER:A: Party during weekdaysB: Going to graduate schoolP(A) = 0.273, P(B | A) = 0.67P(A B) = P(B | A) P(A) = (0.67) (0.273) = 0.1829
QUESTIONS 161 THROUGH 164 ARE BASED ON THE FOLLOWING INFORMATION:A recent marketing survey tried to relate a consumer’s awareness of a new marketing campaign with their rating of the product. Consumers rated their awareness as low, medium, or high, and rated the product as poor, fair, or good. The results are presented below.
161. What is the probability that a consumer had both high awareness and thought the product was poor?
ANSWER:0.07
162. What is the probability that a consumer who had medium awareness ranked the product as fair or good?
164. You can order a new model of a electronic notebook in one of five colors. The notebook also comes loaded with one of four different games. How many different combinations of colors and games are possible? If you were to order three different notebooks without stating preference on color or games, what is the probability that they are identical? Assume that the color and game on one notebook is independent of any others.
ANSWER: There would be (5)(4) =20 possible models. If you choose three different notebooks, the probability the first and the second match is 1/20. The probability the third also matches is 1/20*1/20 = 1/400
QUESTIONS 165 THROUGH 167 ARE BASED ON THE FOLLOWING INFORMATION:A stock analyst has provided estimates of a corporation’s expected return over the next year. This return is likely to depend on the interest rate, so the analyst has developed the following table.
168. As the coordinator for your work group, you are worried about a backlog of 8 projects waiting your attention. You will only be able to get to three of the projects this week. How many ways can you prioritize 3 projects out of the backlog of eight?
ANSWER:There are 8 different projects you could rank first. Given that you have chosen the one ranked first, you have seven possible projects for the second position, then six for the third. Total number of combinations = (8)(7)(6) = 336
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QUESTIONS 169 THROUGH 172 ARE BASED ON THE FOLLOWING INFORMATION:A review of the personnel records of a small corporation has revealed the following information about the number of sick days taken per year and the corresponding probabilities.
Number of Sick Days 0 1 2 3 4 5Probability 0.05 0.22 0.31 0.27 0.13 0.02
Let A be the event that an employee takes more than 2 sick days. Let B be the event that an employee takes less than five sick days.
169. What is the probability of event A?
ANSWER:P(A) = 0.27 + 0.13 + 0.02 = 0.42
170. Are events A and B collectively exhaustive? Why
ANSWER:Yes, since A B =S
171. What is the probability of the complement of A?
ANSWER: P( ) = 1 – P(A) = 1 – 0.42 = 0.58
172. What is the probability of the intersection of events A and B?
ANSWER: P(A B) = P(3) + P(4) = 0.27 + 0.13 = 0.40 QUESTIONS 173 AND 174 ARE BASED ON THE FOLLOWING INFORMATION:The probability that an employee at a company eats lunch at the company cafeteria is 0.23, and the probability that employee is female is 0.52. Assume the probability that an employee eats lunch at the company cafeteria and is female is 0.11.
173. What is the probability that a randomly chosen employee either eats lunch at the company cafeteria or is female?
ANSWER:E = East lunch at cafeteriaF = FemaleSince, P(E) = 0.23, P(F ) = 0.52, and P(E F) = 0.11,
174. Are eating lunch at the company cafeteria and gender of a randomly chosen employee mutually exclusive events?
ANSWER:Clearly eating lunch at the company’s cafeteria and gender of the randomly chosen employee are not mutually exclusive events, since P(E F) = 0.11 0.
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175. The probability that a new small business closes before the end of its first year is 42%. In addition, 37% of all new businesses are started by women. The probability that a new business is either owned by a woman or goes out of business is 62%. Your sister wants to start a new business. What is the probability her business is still open at the end of the first year?
ANSWER:A = A new small business closes before the end of its first year.B = New business started by a woman.
B Total
A 0.17 0.25 0.420.20 0.38 0.58
Total 0.37 0.63 1.00
176. In a survey of top executives, it was found that 17% had traveled internationally on business. The probability of one of these executives fluently speaking a foreign language was found to be 10%. The probability that one of these executives neither spoke a foreign language nor had traveled internationally was 0.81. What is the probability that an executive who speaks a foreign language has traveled internationally?
ANSWER:T =Traveled internationallyS = Speaks a foreign language
S Total
T 0.08 0.09 0.170.02 0.81 0.83
Total 0.10 0.90 1.00
QUESTIONS 177 THROUGH 183 ARE BASED ON THE FOLLOWING INFORMATION:In a survey of top executives, it was found that 46% had either traveled internationally on business or could fluently speak a foreign language. The probability that an executive who had not traveled internationally could speak a foreign language was 10%. It was found that only 4% of the executives had traveled internationally and could speak a foreign language.
177. What is the probability that an executive who speaks a foreign language has not traveled internationally?
ANSWER:T = Travel internationallyS = Speak a foreign language
Since , Hence, P(T)+P(S) = 0.50,
and since P(S) = P(T S) + P( S), then we get P(S) = 0.10 and P(T) = 0.40. Hence,
P( S ) = 0.06, and P( /S) = 0.06/0.10 = 0.60.
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178. What is the probability that an executive who has traveled internationally speaks a foreign language fluently?
184. The probability that a person has an Internet connection at home is 34%. The probability that they have access to the Internet at work is 40%. The probability that a person who has access to the Internet at work also has access at home is 55%. What is the probability that a person with an Internet connection at home also has one at work?
ANSWER:H = Internet at homeW = Internet at workP(H) = 0.34, P(W) = 0.40, P(H | W) = 0.55.Hence, P(H W) =P(H/W) P(W) = (0.55)(0.40) = 0.22, and
P(W | H) = P(W H)/P(H) =0.22/0.34 = 0.647.
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185. There are four copy machines in the library. It is estimated that the probability that a particular copy machine is out of order is 0.3. Assuming that the machines work independently of each other what is the probability that at least one machine is out of order the next time you go to the library?
ANSWER:A = None of the machines is out of order
P(A) = =0.2401
The probability that at least one machine is out of order is P( ) =1-.2401=0.7599 0.76.
186. Firms are increasingly asking applicants to submit to drug tests. Suppose that drug tests can identify a drug user 98% of the time. However, 1% of the time the test indicates a positive result although the applicant is not a drug user. If 10% of all applicants are drug users, what is the probability that a person who has tested positive for drug use is not really a drug user?
ANSWER:D = Drug user+ = Test is positiveSince P(+ | D) = 0.98, P(+ | ) = 0.01, and P(D) = 0.10,
QUESTIONS 187 THROUGH 193 ARE BASED ON THE FOLLOWING INFORMATION:In a recent survey about capital punishment, 64 % of the respondents said that they support capital punishment. Females comprised 48% of the sample, and of the females, 46% supported capital punishment.
187. What is the probability that a randomly selected person is a female and capital punishment supporter?
188. Suppose we select a capital punishment supporter so that we can ask some follow up questions. What is the probability that the person we select is female?
QUESTIONS 194 THROUGH 206 ARE BASEWD ON THE FOLLOWING INFORMATION:Consider a sample space defined by events , Let ) = 0.40, = 0.60,
and = 0.70.
194. What is ?
ANSWER:= 0.40 = 1 – 0.40 = 0.60
195. What is ?
ANSWER:(0.60)(0.40) = 0.24
196. What is ?
ANSWER:0.40 – 0.24 = 0.16
197. What is ?
ANSWER:(0.70)(0.60) = 0.42
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198. What is ?
ANSWER:0.24 + 0.42 = 0.66
199. What is ?
ANSWER: 1 – 0.66 = 0.34
200. What is ?
ANSWER:0.34 – 0.16 = 0.18
201. What is ?
ANSWER:0.24 / 0.66 = 0.3636
202. What is ?
ANSWER: 0.16 / 0.34 = 0.4706
203. What is ?
ANSWER:0.42 / 0.66 = 0.6364
204. What is ?
ANSWER:0.18 / 0.34 = 0.5294
205. What is ?
ANSWER:0.16 / 0.40 = 0.40
206. What is ?
ANSWER:0.18 / 0.60 = 0.30
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QUESTIONS 207 THROUGH 216 ARE BASEWD ON THE FOLLOWING INFORMATION:A publisher sends advertising materials for an economics text to 75% of all professors teaching the appropriate economics course. Twenty eight percent of the professors who received this material adopted the book, as did 8% of the professors who did not receive the material. Define the following events of interest:
= Professor receives advertising material
= Professor does not receive advertising material
= Professor adopts the book
= Professor does not adopt the book
207. What is the probability that a professor who adopts the book has received the advertising material?
ANSWER: = 0.75, = 1 - 0.75 = 0.25, = 0.28, = 0.08
Using Bayes’ Theorem, we get
208. What is the probability that a professor who adopts the book has not received the advertising material?
ANSWER:Using Bayes’ Theorem, we get
Also, since , then 1 – 0.913 = 0.087
209. What is the probability that a professor who receives advertising material has not adopted the book?
ANSWER:, then 1 – 0.28 = 0.72
210. What is the probability that a professor who does not receive advertising material has not adopted the book?
ANSWER:, then 1 – 0.08 = 0.92
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Probability
211. What is the probability that a professor receives advertising material and has adopted the book?
ANSWER:(0.28)(0.75) = 0.21
212. What is the probability that a professor receives advertising material and has not adopted the book?
ANSWER:(0.72)(0.75) = 0.54
213. What is the probability that a professor does not receive advertising material and has adopted the book?
ANSWER:(0.08)(0.25) = 0.02
214. What is the probability that a professor does not receive advertising material and has not adopted the book?
ANSWER:(0.92)(0.25) = 0.23
215. What is the probability that a professor adopts the book?
ANSWER:0.21 + 0.02 = 0.23
216. What is the probability that a professor does not adopt the book?
ANSWER:0.54 + 0.23 = 0.77
Also, since , then 1 – 0.23 = 0.77
217. How would you explain to your boyfriend or girlfriend (I know they are intelligent people!) who have not studied probability at all the difference between mutually exclusive events and independent events? Illustrate your answer with easy to understand examples.
ANSWER:Mutually exclusive events are events such that if one event occurs, the other event cannot occur. For example, a U.S. Senator voting in favor of a tax cut cannot also vote against it. Independent events are events such that the occurrence of one event has no effect on the probability of the other event. For example, whether or not you ate breakfast this morning is unlikely to have any effect on the probability of a U.S. Senator voting in favor of a tax cut.
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QUESTIONS 218 THROUGH 222 ARE BASED ON THE FOLLOWING INFORMATION:A market research firm regularly assembles panel of consumers to test new television commercials for effectiveness. The consumers are told that they are evaluating a pilot TV program. After viewing the hour-long program, complete with commercials, they are asked many questions about the program and some about the commercial – the actual object of research. A tabulation of results from one panel counted the number of panelists who recalled the product incorrectly, the number who recalled the product correctly and had a favorable opinion, and the number who recalled the product correctly and had an unfavorable opinion.
218. Use the addition law to find the probability that a randomly chosen consumer recalled the commercial.
ANSWER:A customer recalling the commercial at all is placed in either the favorable or the unfavorable category. Therefore,P(recalled) = P(favorable) + P(unfavorable) = (95 / 250) + (50 / 250) = 0.58
219. Use the complements principle to find the same probability in Question 218.
ANSWER:A customer who does not recall the commercial is placed in the incorrect category. Therefore, P(incorrect) = 105 / 250 = .42.Hence, P(recalled) = 1 – P(incorrect) = 1 - 0.42 = 0.58
220. What is the probability that a randomly chosen consumer is either a man or someone who recalled the product favorably?
ANSWER:We use the general addition rule, P(man or favorable) = P(man) + P(favorable) – P(man and favorable)
= (100 / 250) + (95 / 250) – (38 / 250) = 0.628
221. Calculate the conditional probabilities of incorrect, favorable, and unfavorable responses among men. Do the same for women.
ANSWER:The conditional probabilities for men may be found by dividing the response frequencies in the “men” row by the total number of men, 100. For women, divide the frequencies by the total number of women, 150. The results are shown in the next table.
222. Are there gender differences in response to the commercial?
ANSWER:The response probabilities in Question 221 are exactly the same for the two groups. Therefore there are no gender differences at all in the responses.
QUESTIONS 223 THROUGH 230 ARE BASED ON THE FOLLOWING INFORMATION:A personnel officer for a firm that employs many part time salespeople tries out a sales aptitude test on several hundred applicants. Because the test is unproven, results are not used in hiring. Forty percent of applicants show high aptitude on the test and 12% of those hired both show high and achieve good sales records. The firm’s experience shows that 30% of all salespeople achieve good sales. Let A be the event “shows high aptitude” and let B be the event “achieves good sales”.
223. What is P(A)?
ANSWER:P(A) = 0.40
224. What is P(A B)?
ANSWER:P(A B) = 0.12
225. What is P(B | A)?
ANSWER:P(B | A) = P(B A) / P(A) = 0.12 / 0.40 = 0.30
226. Are A and B independent?
ANSWER:Since, P(B | A) = P(B) = 0.30, A and B are statistically independent. Alternatively, P(A B) = 0.12 and P(A)P(B) = (0.4)(0.3) = 0.12.
Because P(A B) = P(A) P(B), once again A and B are independent.
227. How useful is the test in predicting good sales achievement?
ANSWER:Because A and B are statistically independent (i.e., the occurrence of event A does not change the probability that B occurs), the test is not at all useful in predicting good sales achievement. An applicant has the same probability of good sales results, regardless of whether or not the applicant showed high aptitude.
228. What is P(A )?
ANSWER:Since P(A) = P(A B) + P(A ), then P(A ) = 0.40 - 0.12 = 0.28
229. What is P( | A)?
ANSWER:P( | A) = P( A) / P(A) = 0.28 / 0.40 = 0.70.
230. Are A and independent?
ANSWER:Since P( | A) = .70, and P( ) = 1 - 0.30 = 0.70, therefore, P( | A) = P( ) = 0.70.
Hence A and are also statistically independent.
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QUESTIONS 231 THROUGH 239 ARE BASED ONTHE FOLLOWING INFORMATION:An investment firm has classified its clients according to their gender and the composition of their investment portfolio (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The proportions of clients falling into the various categories are shown in the following table:
One client is selected at random, and two events A and B are defined as follows:A: The client selected is male.B: The client selected has a balanced portfolio.
231. What is P(A)?
ANSWER:P(A) = 0.18 + 0.20 + 0.25 = 0.63
232 What is P(B)?
ANSWER:P(B) = 0.25 + 0.15 = 0.40
233. Express the event “A B” in words
ANSWER:A B = The client selected either is male or has a balanced portfolio or both.
234. Express the event “A B” in words
ANSWER:A B = The client selected is male and has a balanced portfolio.
235. What is P(A B)?
ANSWER:P(A B) = 0.25
236. Express the probability P(A | B) in words.
ANSWER:P(A | B) = The probability that the employee selected is male, given that the employee has a balanced portfolio.
237. Express the probability P(B | A) in words
ANSWER:P(B | A) = The probability that the employee selected has a balanced portfolio, given that the employee is male.
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238. What is P(A | B)?
ANSWER:P(A | B) = P(A B) / P(B) = 0.25 / 0.40 = 0.625
239. What is P(B | A)?
ANSWER:P(B | A) = P(A B) / P(A) = 0.25 / 0.63 = 0.3968
FOR QUESTIONS 240 THROUGH 249, USE THE FOLLOWING INFORMATION:A table of joint probabilities is shown below, regarding the instructors at the University of
Michigan, where the events (I = 1, 2, 3) and (j = 1, 2) are defined as follows: = Full
professor, = Associate professor, = Assistant professor, = Male instructor, and = Female instructor.
0.15 0.25 0.20
0.10 0.15 0.15
240. Calculate the marginal probabilities of event A.
ANSWER: P( ) = 0.25, P( ) = 0.40, P( ) = 0.35
241. Calculate the marginal probabilities of event B.
ANSWER:P( | ) = 0.4167, and P( ) = 0.40. Since P( | ) P( ), we conclude that the two events are dependent.
QUESTIONS 250 THROUGH 259 ARE BASED ON THE FOLLOWING INFORMATION:Three airlines serve a small town in Indiana. Airline A has 60% of all the scheduled flights, airline B has 30%, and airline C has the remaining 10%. Their on-time rates are 80%, 60%, and 40% respectively. Define event D as an airline arrives on time.
QUESTIONS 260 THROUGH 264 ARE BASED ON THE FOLLOWING INFORMATION:A general contractor has submitted two bids for two projects; A and B. The probability of getting project A is 0.60. The probability of getting project B is 0.75. The probability of getting at least one of the projects is 0.85.
260. What is the probability that the contractor will get both projects?
ANSWER:P(A) = 0.60, P(B) = 0.75, and P(A B) = 0.85
P(A B) = P(A) + P(B) – P(A B) implies that 0.85 = 0.60 + 0.75 - P(A B)
Hence, P(A B) = 0.50
261. Are the events of getting the two projects mutually exclusive? Explain using probabilities.
ANSWER:Since P(A B) = 0.50 0, the two events are not mutually exclusive.
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262. Are the events of getting the two projects independent? Explain using probabilities.
ANSWER:Since P(A) P(B) = (0.60)(0.75) = 0.45 P(A B) = 0.50, the events of getting the two projects are not independent,
263. If the contractor gets project A, what is the probability that he will get project B?
ANSWER:= 0.50 / 0.60 = 0.833
264. If the contractor gets project A, what is the probability that he will get project B?
