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