Chapter 3 Probability True/False 1. A contingency table is a tabular summary of probabilities concerning two sets of complementary events. Answer: True Difficulty: Medium 2. An event is a collection of sample space outcomes. Answer: True Difficulty: Easy 3. Two events are independent if the probability of one event is influenced by whether or not the other event occurs. Answer: False Difficulty: Medium 4. Mutually exclusive events have a nonempty intersection. Answer: False Difficulty: Medium (REF) 5. A subjective probability is a probability assessment that is based on experience, intuitive judgment, or expertise. Answer: True Difficulty: Medium 6. The probability of an event is the sum of the probabilities of the sample space outcomes that correspond to the event. Answer: True Difficulty: Medium 7. If events A and B are mutually exclusive, then P( A B ) is always equal to zero. Answer: True Difficulty: Hard (REF) 8. If events A and B are independent, then P(A|B) is always equal to zero. Answer: False Difficulty: Medium (REF) 9. If events A and B are mutually exclusive, then P(A B) is always equal to zero. Answer: True Difficulty: Easy
10. Events that have no sample space outcomes in common, and, therefore cannot occur simultaneously are referred to as independent events. Answer: False Difficulty: Medium Multiple Choice 11. Two mutually exclusive events having positive probabilities are ______________ dependent. A) Always B) Sometimes C) Never Answer: A Difficulty: Hard (REF) 12. ___________________ is a measure of the chance that an uncertain event will occur. A) Random experiment B) Sample Space C) Probability D) A complement E) A population Answer: C Difficulty: Medium 13. A manager has just received the expense checks for six of her employees. She randomly distributes the checks to the six employees. What is the probability that exactly five of them will receive the correct checks (checks with the correct names). A) 1 B) ½ C) 1/6 D) 0 E) 1/3 Answer: D Difficulty: Hard 14. In which of the following are the two events A and B, always independent? A) A and B are mutually exclusive. B) The probability of event A is not influenced by the probability of event B. C) The intersection of A and B is zero. D) P(A/B) = P(A). E) B and D. Answer: E Difficulty: Hard (REF)
15. If two events are independent, we can _____ their probabilities to determine the intersection probability. A) Divide B) Add C) Multiply D) Subtract Answer: C Difficulty: Easy 16. Events that have no sample space outcomes in common, and therefore, cannot occur simultaneously are: A) Independent B) Mutually Exclusive C) Intersections D) Unions Answer: B Difficulty: Medium 17. If events A and B are independent, then the probability of simultaneous occurrence of event A and event B can be found with: A) P(A)P(B) B) P(A)P( B A ) C) P(B)P( A B ) D) All of the above are correct Answer: D Difficulty: Hard (REF) 18. The set of all possible experimental outcomes is called a(n): A) Sample space B) Event C) Experiment D) Probability Answer: A Difficulty: Easy 19. A(n) ____________ is the probability that one event will occur given that we know that another event already has occurred. A) Sample space outcome B) Subjective Probability C) Complement of events D) Long-run relative frequency E) Conditional probability Answer: E Difficulty: Medium
20. The _______ of two events X and Y is another event that consists of the sample space outcomes belonging to either event X or event Y or both event X and Y. A) Complement B) Union C) Intersection D) Conditional probability Answer: B Difficulty: Medium 21. If P(A) > 0 and P(B) > 0 and events A and B are independent, then: A) P(A) = P(B) B) P( A B )=P(A) C) P(A B) = 0 D) P(A B) = P(A) P(B A) Answer: B Difficulty: Medium 22. P(A B) = P(A) + P(B) - P(A B) represents the formula for the A) conditional probability B) addition rule C) addition rule for two mutually exclusive events D) multiplication rule Answer: B Difficulty: Medium 23. The management of a company believes that weather conditions significantly affect the level of demand for its product. 48 monthly sales reports are randomly selected. These monthly sales reports showed 15 months with high demand, 28 months with medium demand, and 5 months with low demand. 12 of the 15 months with high demand had favorable weather conditions. 14 of the 28 months with medium demand had favorable weather conditions. Only 1 of the 5 months with low demand had favorable weather conditions. What is the probability that weather conditions are poor, given that the demand is high? A) .2 B) .5 C) .8 D) .25 E) .75 Answer: A Difficulty: Hard
24. The management believes that the weather conditions significantly impact the level of demand and the estimated probabilities of poor weather conditions given different levels of demand is presented below. P ( Poor High) .2, P ( Poor Medium) .5, P ( Poor Low) .8 What is the probability of high demand given that the weather conditions are poor. A) .06 B) .44 C) .1364 D) .12 E) .1818 Answer: C Difficulty: Hard Use the following information to answer questions 25-26: An automobile insurance company is in the process of reviewing its policies. Currently drivers under the age of 25 have to pay a premium. The company is considering increasing the value of the premium charged to drivers under 25. According to company records, 35% of the insured drivers are under the age of 25. The company records also show that 280 of the 700 insured drivers under the age of 25 had been involved in at least one automobile accident. On the other hand, only 130 of the 1300 insured drivers 25 years or older had been involved in at least one automobile accident. 25. An accident has just been reported. What is the probability that the insured driver is under the age of 25? A) 35% B) 20.5% C) 14% D) 68.3% E) 40% Answer: D Difficulty: Hard (AS) 26. What is the probability that an insured driver of any age will be involved in an accident? A) 35% B) 20.5% C) 65% D) 68.3% E) 79.5% Answer: B Difficulty: Hard (AS)
27. A pharmaceutical company manufacturing pregnancy test kits wants to determine the probability of a woman not being pregnant when the test results indicate pregnancy. It is estimated that the probability of pregnancy among potential users of the kit is 10%. According to the company laboratory test results 1 out of 100 non-pregnant women tested pregnant (false positive). On the other hand, 1 out of 200 pregnant women tested nonpregnant (false negative). A woman has just used the pregnancy test kit manufactured by the company and the results showed pregnancy. What is the probability that she is not pregnant? A) 90% B) 0.9% C) 8.3% D) 91.7% E) 10.85% Answer: C Difficulty: Hard 28. A pharmaceutical company manufacturing pregnancy test kits wants to determine the probability of a woman actually being pregnant when the test results indicate that she is not pregnant. It is estimated that the probability of pregnancy among potential users of the kit is 10%. According to the company laboratory test results 1 out of 100 nonpregnant women tested pregnant (false positive). On the other hand, 1 out of 200 pregnant women tested non-pregnant (false negative). A woman has just used the pregnancy test kit manufactured by the company and the results showed that she is not pregnant. What is the probability that she is pregnant? A) 1% B) 0.9% C) 0.05% D) 8.3% E) 0.056% Answer: E Difficulty: Hard Fill-in-the-Blank 29. A(n) _____ is the set of all of the distinct possible outcomes of an experiment. Answer: Sample Space Difficulty: Medium 30. The _____ of an event is a number that measures the likelihood that an event will occur when an experiment is carried out. Answer: Probability Difficulty: Easy 31. When the probability of one event is influenced by whether or not another event occurs, the events are said to be _____.
Answer: Dependent
Difficulty: Medium
32. A process of observation that has an uncertain outcome is referred to as a(n) _____. Answer: Experiment Difficulty: Medium 33. When the probability of one event is not influenced by whether or not another event occurs, the events are said to be _____. Answer: Independent Difficulty: Medium 34. A probability may be interpreted as a long run _____ frequency. Answer: Relative Difficulty: Medium 35. If events A and B are independent, then P(A/B) is equal to _____. Answer: P(A) Difficulty: Medium 36. The simultaneous occurrence of event A and B is represented by the notation: _______. Answer: A B Difficulty: Easy 37. A(n) _______________ probability is a probability assessment that is based on experience, intuitive judgment, or expertise. Answer: Subjective Difficulty: Medium 38. A(n) ______________ is a collection of sample space outcomes. Answer: Event Difficulty: Easy 39. Probabilities must be assigned to experimental outcomes so that the probabilities of all the experimental outcomes must add up to ___. Answer: 1 Difficulty: Easy 40. Probabilities must be assigned to experimental outcomes so that the probability assigned to each experimental outcome must be between ____ and ____ inclusive. Answer: 0,1 Difficulty: Easy 41. The __________ of event X consists of all sample space outcomes that do not correspond to the occurrence of event X. Answer: Complement Difficulty: Easy
42. The _______ of two events A and B is another event that consists of the sample space outcomes belonging to either event A or event B or both event A and B. Answer: Union Difficulty: Easy 43. The _______ of two events A and B is the event that consists of the sample space outcomes belonging to both event A and event B. Answer: Intersection Difficulty: Easy 44. __________________ statistics is an area of statistics that uses Bayes' theorem to update prior belief about a probability or population parameter to a posterior belief. Answer: Bayesian Difficulty: Medium 45. In the application of Bayes' theorem the sample information is combined with prior probabilities to obtain ___________________ probabilities. Answer: posterior Difficulty: Easy (REF) Essay 46. What is the probability of rolling a seven with a pair of fair dice? Answer: 1/6 6 36 Difficulty: Medium 47. What is the probability of rolling a value higher than eight with a pair of fair dice? Answer: .2777 10 .2777 36 Difficulty: Medium 48. What is the probability that an even number appears on the toss of a die? Answer: .5 Difficulty: Easy 49. What is the probability that a king appears in drawing a single card form a deck of 52 cards? Answer: 1/13 Difficulty: Medium
50. If we consider the toss of four coins as an experiment, how many outcomes does the sample space consist of? Answer: 16 2 4 16 Difficulty: Medium 51. What is the probability of at least one tail in the toss of three fair coins? Answer: 7/8 Difficulty: Hard 52. A lot contains 12 items, and 4 are defective. If three items are drawn at random from the lot, what is the probability they are not defective? Answer: .2545 8 7 6 .2545 12 11 10 Difficulty: Hard
53. A person is dealt 5 cards from a deck of 52 cards. What is the probability they are all clubs? Answer: .0004951 13 12 11 52 51 50 Difficulty: Hard
10 49
9 0.0004951 48
54. A group has 12 men and 4 women. If 3 people are selected at random from the group, what is the probability that they are all men? Answer: .392857 12 11 10 .392857 15 16 14 Difficulty: Hard
Use the following information to answer questions 55-57: Container 1 has 8 items, 3 of which are defective. Container 2 has 5 items, 2 of which are defective. If one item is drawn from each container:
55. What is the probability that both items are not defective? Answer: .375
5 3 .375 8 5
Difficulty: Medium
56. What is the probability that the item from container one is defective and the item from container 2 is not defective? Answer: .225 3 3 .225 8 5
Difficulty: Hard 57. What is the probability that one of the items is defective? Answer: .45 3 3 3 8 5 8
3 .45 5
Difficulty: Hard 58. A coin is tossed 6 times. What is the probability that at least one head occurs? Answer: 63/64 1 1 1 1 2 2 2 Difficulty: Medium
1 2
1 2
1 2
63 64
59. Suppose P(A) = .45, P(B) =.20, P(C) = .35, P( E A ) = .10, P( E B ) = .05, and P( E C ) = 0. What is P(E)? Answer: .055 P(E) = (.45)(.10) + (.20)(.05) + (.35)(0) = .055 Difficulty: Hard
60. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P( E A ) = .10, P( E B ) = .05, and P( E C ) = 0. What is P( A E )? Answer: .8182 P A E (.10)(.45) .045 P( E ) (.45)(.10) (.20)(.05) (.35)(0) .055 P( A E )
P( A E ) .045 .8182 P( E ) .055
Difficulty: Hard 61. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P( E A ) = .10, P( E B ) = .05, and P( E C ) = 0. What is P( B E )? Answer: 1818 P B E (.20)(.05) .01 P( E ) (.45)(.10) (.20)(.05) (.35)(0) .055 P( B E )
P( B E ) .01 .1818 P( E ) .055
Difficulty: Hard 62. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P( E A ) = .10, P( E B ) = .05, and P( E C ) = 0. What is P ( C E )? Answer: 0 P C E 0 P( E ) (.45)(.10) (.20)(.05) (.35)(0) .055 P(C E )
P(C E ) 0 0 P( E ) .055
Difficulty: Hard 63. Given the standard deck of cards, what is the probability of drawing a red card, given that it is a face card? Answer: .5 6 P (Re d Face) P (Re d Face) 52 .5 12 P ( Face) 52 Difficulty: Medium
64. Given a standard deck of cards, what is the probability of drawing a face card, given that it is a red card? Answer: 3/13 6 P (Re d Face) 3 P ( Face Re d ) 52 26 P (Re d ) 13 52 Difficulty: Medium 65. A machine is made up of 3 components: an upper part, a mid part, and a lower part. The machine is then assembled. 5 percent of the upper parts are defective; 4 percent of the mid parts are defective; 1 percent of the lower parts are defective. What is the probability that a machine is non-defective? Answer: .9029 (.95)(.96)(.99) = .9029 Difficulty: Hard (AS) 66. A machine is produced by a sequence of operations. Typically one defective machine is produced per 1000 parts. What is the probability of two non-defective machines being produced? Answer: .998 (.999)(.999) = .998 Difficulty: Medium 67. A pair of dice is thrown. What is the probability that one of the faces is a 3, given that the sum of the two faces is 9? Answer: 1/4 Difficulty: Hard 68. A card is drawn from a standard deck. What is the probability the card is an ace, given that it is a club? Answer: 1/13 Difficulty: Medium 69. A card is drawn from a standard deck. Given that a face card is drawn, what is the probability it will be a king? Answer: 1/3 (4 kings)/(12 face cards) Difficulty: Medium
70. Independently a coin is tossed, a card is drawn from a deck, and a die is thrown. What is the probability of observing a head on the coin, an ace on the card, and a five on the die? Answer: 1/156 1 1 4 1 52 156 2 6 Difficulty: Medium
71. A family has two children. What is the probability that both are girls, given that at least one is a girl? Answer: 1/3 Difficulty: Medium 72. What is the probability of winning four games in a row, if the probability of winning each game individually is 1/2? Answer: 1/16 Difficulty: Medium Use the following to answer questions 73-77: At a college, 70 percent of the students are women and 50 percent of the students receive a grade of C. 25 percent of the students are neither female nor C students. Use this contingency table.
Women Men
C .45 .05 .50
C .25 .25 .50
.70 .30 1.00
73. What is the probability that a student is female and a C student? Answer: .45 Difficulty: Hard (AS) 74. What is the probability that a student is male and not a C student? Answer: .25 Difficulty: Hard (AS)
75. If the student is male, what is the probability he is a C student? Answer: .1667 P (C Male)
.05 .1667 .30
Refer To: 03_02 Difficulty: Hard (AS) 76. If the student has received a grade of C, what is the probability that he is male? Answer: .10 .05 P ( Male C ) .10 .50 Difficulty: Hard
(AS)
77. If the student has received a grade of C, what is the probability that she is female? Answer: .90 .45 P ( female C ) .90 .50 Difficulty: Hard
(AS)
Use the following information to answer questions 78-79: Two percent (2%) of the customers of a store buy cigars. Half of the customers who buy cigars buy beer. 25 percent who buy beer buy cigars. Determine the probability that a customer using this contingency table:
Cigars Cigars
Beer .01 .03
Beer .01 .95
.02 .95
.04
.96
1.0
78. Buys beer. Answer: .04 P ( Beer Cigar ) .01 P ( Beer ) .04 P ( Beer Cigar ) .25 Difficulty: Hard
79. Neither buys beer nor buys cigars. Answer: .95 P ( Beer Cigar ) .01 P ( Beer ) .04 P( Beer Cigar ) .25 P ( No Cigar Beer ) .03 P ( No Cigar No Beer ) .98 .03 .95
Difficulty: Hard Use the following information to answer questions 80-81: An urn contains five white, three red, and four black balls. Three are drawn at random without replacement. 80. What is the probability that no ball is red? Answer: .3818 9 8 7 .3818 12 11 10 Difficulty: Hard
81. What is the probability that all balls are the same color? Answer: .0682 C35 C33 C34 10 1 4 .0682 C312 220 Difficulty: Hard 82. What is the probability that any two people chosen at random were born on the same day of the week? Answer: 1/7 Difficulty: Hard 83. A letter is drawn from the alphabet of 26 letters. What is the probability that the letter drawn is a vowel? Answer: 5/26 Difficulty: Easy 84. How many times must a die be tossed if the expected number of ones is five? Answer: 30 Difficulty: Medium
85. List two properties of a valid discrete probability distribution. Answer: n
p ( X ) 0 , for all X and X i 1 i
Difficulty: Hard 86. If A and B are independent events, P(A) = .2, and P(B) = .7, determine P(A B) Answer: .76 P(A B) = P(A) + P(B) – P(A B) P(A B) = (.7) + (.2) – (.7)(.2) = .76 Difficulty: Medium 87. If events A and B are mutually exclusive, calculate P( A B ). Answer: Zero Difficulty: Hard 88. What is the probability of rolling a six with a fair die five times in a row? Answer: 1/7,776 1 1 1 1 6 6 6 6 Difficulty: Hard
1 6
1 1 6 7776
89. If a product is made using five individual components, and P(product meets specifications) = .98, what is the probability of an individual component meeting specifications assuming that this probability is the same for all five components? Answer: .9960 5
.98 Difficulty: Hard
(AS)
90. If P( A B ) = .2 and P(B) = .8, determine the intersection of event A and B. Answer: .16 (.2)(.8) = .16 Difficulty: Medium
91. If P(A B )= .3 and P( A B ) = .9, find P(B). Answer: .333 .3 P ( B ) .333 .9 Difficulty: Medium Use the following information to answer questions 92-100: Job Faculty (FA) Salaried staff (SS) Hourly staff (HS)
Gender Male (M) Female (F) 110 10 30 50 60 40
Employees of a local university have been classified according to gender and job type. 92. If an employee is selected at random what is the probability that the employee is male? Answer: .667 200 P(M ) .667 300 Difficulty: Medium (AS) 93. If an employee is selected at random what is the probability that the employee is male and salaried staff? Answer: 0.10 30 P ( M and SS ) 0.10 300 Difficulty: Medium (AS) 94. If an employee is selected at random what is the probability that the employee is female given that the employee is a salaried member of staff? Answer: 0.625 50 5 P ( F SS ) 0.625 8 80 Difficulty: Medium (AS)
95. If an employee is selected at random what is the probability that the employee is female or works as a member of the faculty? Answer: 0.70 100 120 10 P ( F FA) 0.70 300 300 300 Difficulty: Medium (AS)
96. If an employee is selected at random what is the probability that the employee is female or works as an hourly staff member? Answer: 0.533 100 100 40 P ( F HA) 0.533 300 300 300 Difficulty: Medium (AS) 97. If an employee is selected at random what is the probability that the employee is a member of the hourly staff given that the employee is female? Answer: 0.40 40 4 P ( HS F ) 0.40 10 100 Difficulty: Medium (AS) 98. If an employee is selected at random what is the probability that the employee is a member of the faculty? Answer: .40 120 P( FA) .40 300 Difficulty: Medium (AS) 99. Is gender and type of job mutually exclusive? Explain with probabilities. Answer: No, gender and job type is not mutually exclusive. 200 .667 300 120 P ( FA) 0.40 300 110 P ( M FA) 0 300 Difficulty: Medium (AS) P(M )
100. Is gender and type of job statistically independent? Explain with probabilities. Answer: No, gender is not independent of type of job. Select a category of gender (male) and a category of job status (faculty), if the two are independent of each other, than: P ( M ) P( M FA) Since P ( M ) 200
0.667 300 and P ( M FA) 110 0.9167 120 0.667 .9167 Difficulty: Medium (AS)
Use the following information to answer questions 101-104: Worker Joe Jan Cheryl Clay
% of Dinners Packed 25% 20% 20% 35%
% Forgot Napkin 6% 2% 10% 4%
Four employees who work as drive-through attendees at a local fast food restaurant are being evaluated. As a part of quality improvement initiative and employee evaluation these workers were observed over three days. One of the statistics collected is the proportion of time employee forgets to include a napkin in the bag. Related information is given in the table, above. 101. What is the probability that Cheryl prepared your dinner and forgot to include a napkin? Answer: 0.02 P(Cheryl Forgot napkin) = (.20)(.10) =0.02 Difficulty: Medium Refer To: 03_06 102. What is the probability that there is not a napkin included for a given order? Answer: 0.053 P(No Napkin) = (.25)(.06) + (.20)(.02) + (0.20)(.10) + (.35)(.04) = .053 Difficulty: Medium
103. You just purchased a dinner and found that there is no napkin in your bag, what is the probability that Cheryl has prepared your order? Answer: 0.3774 (.2)(.1) .02 P (Cheryl No napkin) 0.37774 (.25)(.06) (.2)(.02) (.20)(.10) (.35)(.04) .053 Difficulty: Hard 104. You just purchased a dinner and found that there is no napkin in your bag, what is the probability that Jan has prepared your order. Answer: 0.0754 (.2)(.02) .004 P ( Jan No napkin) 0.0755 (.25)(.06) (.2)(.02) (.20)(.10) (.35)(.04) .053 Difficulty: Hard Use the following information to answer questions 105-110: Joe is considering pursuing an MBA degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 25% for University A and 40% for University B. 105. What is the probability that Joe will be accepted at both universities? Answer: 0.10 (.25)(.40) = 0.10 Difficulty: Hard 106. What is the probability that Joe will be accepted at University A and rejected at University B? Answer: 0.15 (.25)(.60) = 0.15 Difficulty: Medium 107. What is the probability that Joe will not be accepted at either university? Answer: 0.45 (.75)(.60) = 0.45 Difficulty: Medium 108. What is the probability that Joe will be accepted at least by one of the two universities? Answer: 0.55 1- [(.75)(.60)] = 0.55 Difficulty: Medium
109. What is the probability that Joe will be accepted at one, and only one university? Answer: 0.45 (.25)(.60) + (.75)(.40) = 0.45 Difficulty: Hard 110. Is the acceptance decision at University A independent of the acceptance decision at University B? Show with probabilities. Answer: Yes, the two decisions are statistically independent If the MBA acceptance decisions are independent at the two universities, than P(Accepting at A) = P(Accepting at A given rejecting at B). P( Accept at A) .25 P( Accept at A Re ject at B )
(.25)(.60) .15 .25 .60 .60
Difficulty: Hard Use the following information to answer questions 111-113: In a report on high school graduation, it was stated that 85% of high school students graduate. Suppose 3 high school students are randomly selected from different schools. 111. What is the probability that all graduate? Answer: 0.614 (85)(.85)(.85) = 0.614 Difficulty: Easy 112. What is the probability that exactly one of the three graduate? Answer: .0574 (.85)(.15)(.15)+(.15)(.85)(.15)+(.15)(.15)(.85)= .057375 Difficulty: Hard 113. What is the probability that none graduate? Answer: .0034 (.15)(.15)(.15) = .003375 Difficulty: Easy Use the following information to answer questions 114-116: It is very common for television series to draw a large audience for special events of for cliff-hanging story lines. Suppose that on one of these occasions, the special show drew viewers from 38.2% of all US TV households. Suppose that three TV households are randomly selected.
114. What is the probability that all three households viewed this special show? Answer: .056 (.382)(.382)(.382) = .05574 Difficulty: Easy 115. What is the probability that none of the three households viewed this special show? Answer: .236 (.618)(.618)(.618)=.236 Difficulty: Easy 116. What is the probability that exactly one of the three households viewed the special show? Answer: .438 (.382)(.618)(.618)+(.618)(.382)(.618)+(.618)(.618)(.382)=.4376 Difficulty: Hard Use the following information to answer questions 117-119: A survey is made in a neighborhood of 80 voters. 65 were Democrats and 15 were Republicans. Of the Democrats, 35 are women, while 5 of the Republicans are women. If one subject from the group is randomly selected, find the probability: 117. The individual is either a woman or a Democrat. Answer: .875 P(W D) = P(W) + P(D) – P(W D) (40/80) + (65/80) – (35/80) = .875 Difficulty: Medium 118. A male Republican Answer: .125 (10/80) = .125 Difficulty: Medium 119. A Democrat of a Republican Answer: 1.00 Difficulty: Easy Use the following information to answer questions 120-121: Owners are asked to evaluate their experiences in buying a new car during the past
twelve months. When surveys were analyzed the owners indicated they were most satisfied with their experiences at the following three dealers (in no particular order): Saturn, Honda, and Buick. 120. List all possible sets of rankings for these three dealers: Answer: HSB, BHS, SHB, HBS, BSH, SBH Difficulty: Easy 121. Assuming that each set of rankings is equally likely, what is the probability that (a) Owners ranked Saturn first? (b) Owners ranked Saturn third? (c) Owners ranked Saturn first and Honda second? Answer: (a) 2/6 (b) 2/6 (c) 1/6 Difficulty: Easy 122. In a study of chain saw injuries, 57% involved arms or hands. If three different chain saw injury cases are randomly selected, find the probability that they all involved arms or hands? Answer: .185 (.57)(.57)(.57)= .185 Difficulty: Easy Use the following information to answer questions 123-125: In a local survey, 100 citizens indicated their opinions on a revision to a local land use plan. Of the 62 favorable responses, there were 40 males. Of the 38 unfavorable responses, there were 15 males. If one citizen is randomly selected find the probability 123. A female or has an unfavorable opinion Answer: .60 P(F N(F) + P(N) – P(F N) .45 + .38 - .23 = .60 Y N M 40 15 65 F 22 23 45 62 38 100 Difficulty: Medium
124. A male has a favorable opinion Answer: .40 Difficulty: Easy 125: Has a favorable opinion or has an unfavorable opinion Answer: 1.00 Difficulty: Easy Use the following information to answer questions 126-130. Determine whether the two events are mutually exclusive: 126. A consumer with an unlisted phone number and a consumer who does not drive Answer: Not mutually exclusive Difficulty: Easy 127. An unmarried person and a person with an employed spouse Answer: Mutually exclusive Difficulty: Easy 128. Someone born in the United States and a US citizen Answer: Not mutually exclusive Difficulty: Easy 129. A voter who favors gun control and a conservative voter Answer: Not mutually exclusive Difficulty: Easy 130. A voter who is a registered Democrat and a voter who favors a Republican candidate Answer: Not mutually exclusive Difficulty: Easy Use the following information to answer questions 131-134: In a recent survey of homes in a major Midwestern city, 10% of the homes have a fax machine and 52% have a personal computer. Suppose 91% of the homes with a fax machine also have a personal computer.
131. What percent of homes have a fax machine and a personal computer? Answer: .091 = 9.1% Difficulty: Hard
132. What is the probability that a home has a fax machine or a personal computer? Answer: .53= 53% P(PC)+P(F)-P(F PC)= (.52)+(.10)-(.09)=.53 Difficulty: Hard 133. What is the probability that a home with a personal computer has a fax machine? Answer: .17 P(F|PC) = P(F PC)/P(PC) = .09/.52= .17 Difficulty: Hard 134. Are the events “owning a fax machine” and “owning a personal computer” independent? Why or why not? Answer: NO P(PC|F) P(PC) Difficulty: Hard 135. A batch of 50 parts contains 6 defects. If two parts are drawn randomly one at a time without replacement, what is the probability that both parts are defective? Answer: .012 (6/50)(5/49)= .012 Difficulty: Medium 136. A batch of 50 parts contains 6 defects. If two parts are drawn randomly, one at a time with replacement, what is the probability that both parts are defective? Answer: .014 (6/50)(6/50)= .014 Difficulty: Medium 137. In the word BUSINESS, what is the probability of randomly selecting the letter S? Answer: .375 3/8= .375 Difficulty: Easy 138. Suppose that you believe that the probability you will get a grade of B or better in Introduction to Finance is .6, and the probability that you will get a grade of B or better in Introduction to Accounting is .5. If these events are independent, what is the probability that you will be a grade of B or better in both courses? Answer: .30 (.6)(.5)=.30 Difficulty: Easy (AS)
Use the following information to answer questions 139-142: In a major Midwestern university, 55% of all undergraduates are female, 25% belong to a Greek organization (fraternity or sorority) and 40% of all males belong to a Greek organization. 139. What percent of the undergraduates are female and in a Greek organization? Answer: 7% Female Male Greek .07 .18 .25 Non-Greek .48 .27 .75 .55 .45 Difficulty: Hard 140. What is the probability that one randomly selected undergraduate will be either a female or belong to a Greek organization? Answer: 73% P(G F) = .55 + .25 -.07=.73 Difficulty: Hard 141. What is the probability that an undergraduate is in a Greek organization given that the undergraduate is a female? Answer: .127 P(G|F) = .07/.55 = .127 Difficulty: Hard 142. Are the events “female/not female” and “belongs to a Greek organization” independent? Answer: No P(G|F) P(G) Difficulty: Hard Use the following information to answer questions 143-146: At a certain university, 30% of the students major in zoology. Of the students majoring in zoology, 60% are males. Of all the students at the university, 70% are males.
143. What percentage of the students are males majoring in zoology? Answer: 18% Zoology Not Major Zoology Male .18 .52 .70 Female .12 .18 .30 .30 .70 1.00 Difficulty: Hard 144. What is the probability that one randomly selected student is a male or is majoring in zoology? Answer: 82% (.30)+(.70)-(.18)=.82 Difficulty: Hard 145. What proportion of the males are majoring in zoology? Answer:.257 P(Z|M) = .18/.70= .257 Difficulty: Hard 146. Are the events “male” and “majoring in zoology” independent? Answer: No P(M|Z) ≠P(M) Difficulty: Hard 147. An advertising campaign is being developed to promote a new bookstore opening in the newest mall development. To develop an appropriate mailing list it has been decided to purchase lists of credit card holders from MasterCard and American Express. Combining the lists they find the following: 40% of the people on the list have only a MasterCard and 10% have only an American Express card. Another 20% hold both MasterCard and American Express. Finally, 30% of those on the list have neither card. Suppose a person on the list is known to have a MasterCard. What is the probability that person also has an American Express Card? Answer: .33 P(AE|MC)=.20/.60=.33 MasterCard No MasterCard American Express .20 .10 .30 No American .40 .30 .70 Express .60 .40 1.00
Difficulty: Medium
Use the following information to answer questions 148-152: Consider a standard deck of 52 playing cards, a randomly selected card from the deck, and the following events: R=red, B=black, A=ace, N=nine, D=diamond and C=club. For each of the following pair of events, indicated whether the events are mutually exclusive 148. R and A Answer: No Difficulty: Easy 149. R and C Answer: Yes Difficulty: Easy 150. A and N Answer: Yes Difficulty: Easy 151. N and C Answer: No Difficulty: Easy 152. D and C Answer: Yes Difficulty: Easy Use the following information to answer questions 153-154: An automobile insurance company is in the process of reviewing its policies. Currently drivers under the age of 25 have to pay a premium. The company is considering increasing the value of the premium charged to drivers under 25. According to company records, 35% of the insured drivers are under the age of 25. The company records also show that 280 of the 700 insured drivers under the age of 25 had been involved in at least one automobile accident. On the other hand, only 130 of the 1300 insured drivers 25 years or older had been involved in at least one automobile accident.
153. An accident has just been reported. What is the probability that the insured driver is under the age of 25? Answer: P ( 25 Acc) .683 State of Nature
P(Sj)
S1 (under 25) S2 (25 or older)
0.35 0.65 1.0 Difficulty: Hard (AS)
P ( Accident S j )
P ( Acc. S j )
P ( S j Acc.)
0.40 0.10
0.14 0.065 0.205
0.683 0.317 1.0
154. What is the probability that an insured driver of any age will be involved in an accident? Answer: P ( Accident ) .205 State of Nature
P(Sj)
S1 (under 25) S2 (25 or older)
0.35 0.65 1.0 Difficulty: Hard (AS)
P ( Accident S j )
P ( Acc. S j )
P ( S j Acc.)
0.40 0.10
0.14 0.065 0.205
0.683 0.317 1.0
155. A pharmaceutical company manufacturing pregnancy test kits wants to determine the probability of a woman not being pregnant when the test results indicate pregnancy. It is estimated that the probability of pregnancy among potential users of the kit is 10%. According to the company laboratory test results, 1 out of 100 non-pregnant women tested pregnant (false positive). On the other hand, 1 out of 200 pregnant women tested non-pregnant (false negative). A woman has just used the pregnancy test kit manufactured by the company and the results showed pregnancy. What is the probability that she is not pregnant? Answer: P( NP TP ) .083 Let: P = pregnant NP = non-pregnant TP = tested pregnant TNP = tested non-pregnant State of Nature
P(Sj)
S1 (pregnant) 0.1 S2 (not pregnant) 0.9
P (TP S j )
P (TP S j )
P ( S j TP)
0.995 0.01
0.0995 0.009
0.917 0.083
1.0 Difficulty: Hard
0.1085
1.0
156. A pharmaceutical company manufacturing pregnancy test kits wants to determine the probability of a woman actually being pregnant when the test results indicate that she is not pregnant. It is estimated that the probability of pregnancy among potential users of the kit is 10%. According to the company laboratory test results, 1 out of 100 nonpregnant women tested pregnant (false positive). On the other hand, 1 out of 200 pregnant women tested non-pregnant (false negative). A woman has just used the pregnancy test kit manufactured by the company and the results showed that she is not pregnant. What is the probability that she is pregnant? Answer: P ( P NTP) .00056 Let: P = pregnant NP = non-pregnant TP = tested pregnant TNP = tested non-pregnant State of Nature
P(Sj)
P (TNP S j )
P (TNP S j )
P ( S j TNP)
S1 (pregnant) S2 (not pregnant)
0.1 0.9 1.0
0.005 0.99
0.0005 0.891 0.8915
0.00056 0.99944 1.0
Difficulty: Hard
