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
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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)
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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
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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
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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)
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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 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?
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 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 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
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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
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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
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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   10   9 





  0.0004951
 52   51   50   49   48 
Difficulty: Hard
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
 16   15   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:
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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  3 
          .45
 8  5   8  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   1   1   63
1               
 2   2   2   2   2   2   64
Difficulty: Medium
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
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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 ) .045

 .8182
P( E )
.055
Difficulty: Hard
P( A E ) 
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 ) .01

 .1818
P( E )
.055
Difficulty: Hard
P( B E ) 
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 )
0

0
P( E )
.055
Difficulty: Hard
P(C E ) 
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
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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
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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
 
  
 2   52   6  156
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)
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75. If the student is male, what is the probability he is a C student?
Answer: .1667
.05
 .1667
.30
Refer To: 03_02
Difficulty: Hard (AS)
P(C Male) 
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
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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
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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  1  1  1 
       
 6   6   6   6   6   6  7776
Difficulty: Hard
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
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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)
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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 ) 
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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
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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
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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 cliffhanging 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.
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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.
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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
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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
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107
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)
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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.
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109
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
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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.
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111
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)
P(TP S j )
P(TP  S j )
P(S j TP)
S1 (pregnant)
S2 (not pregnant)
0.1
0.9
1.0
0.995
0.01
0.0995
0.009
0.1085
0.917
0.083
1.0
Difficulty: Hard
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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 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 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
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113