Chapter 3 – Probability Distributions
True / False
1. Experimental outcomes must occur as numerical values in order to define their probability distribution.
a. True
b. False
2. A random variable that can take on any value along a line segment is a continuous random variable.
a. True
b. False
3. The probability of a continuous variable having a specific value is 0.
a. True
b. False
4. The expected value of the discrete random variable x is μ, a weighted average of all possible values of x.
a. True
b. False
5. If one wanted to find the probability of ten customer arrivals in an hour at a service station, one would generally use the
Poisson distribution.
a. True
b. False
6. The binomial distribution is appropriate to use to find the probability of the elapsed time between successes.
a. True
b. False
7. The Poisson probability of three events in one hour is the same as the probability of 36 events in one day.
a. True
b. False
Chapter 3 – Probability Distributions
8. The probability density function matches each value of a continuous random variable with its probability.
a. True
b. False
9. Suppose that the arrival time for the next bus at a bus stop is uniformly distributed between 1 and 25 minutes. The
probability that the bus will arrive within the next 5 minutes is 1/6.
a. True
b. False
10. If the Poisson distribution describes the number of occurrences per interval, the exponential distribution describes the
length of the interval between occurrences.
a. True
b. False
11. The Poisson probability distribution is used with a continuous random variable.
a. True
b. False
12. Probabilities for discrete random variables are defined over intervals by a probability density function.
a. True
b. False
13. Whenever the probability is proportional to the length of the interval, the random variable is uniformly distributed.
a. True
b. False
Chapter 3 – Probability Distributions
14. A random variable that has a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a
binomial distribution.
a. True
b. False
15. For an exponentially distributed random variable x with a mean of 8, the probability of x ≤ 8 is .5.
a. True
b. False
16. The binomial probability distribution is most symmetric when p equals .5.
a. True
b. False
17. A random variable that may assume only a finite or an infinite sequence (e.g., 1, 2, 3,…) of values is a discrete random
variable.
a. True
b. False
18. An exponential distribution, like the Poisson distribution, can be described by a single parameter.
a. True
b. False
19. The standard normal distribution is a normal distribution with a mean of 1 and a standard deviation of 1/3.
a. True
b. False
20. Weight, time, and temperature are examples of discrete random variables.
a. True
b. False
Chapter 3 – Probability Distributions
21. A numerical description of the outcome of an experiment is
a. a normal variable.
b. a discrete variable.
c. a random variable.
d. an experimental variable.
22. Which of the following are continuous random variables?
I. the weight of an elephant
II. the time to answer a questionnaire
III. the number of floors in a skyscraper
IV. the square feet of countertop in a kitchen
a. I and II only
b. III and IV only
c. I, II and IV
d. I, II, II, and IV
23. A statement that matches the values of a random variable with the probabilities of those values is
a. the expected value.
b. the variation of the random variable.
c. an experiment.
d. a probability distribution.
24. In order to measure the dispersion of a random variable, look at its
a. standard deviation.
b. mean.
c. expected value.
d. average.
25. Experiments with repeated independent trials will be described by the binomial distribution if
a. the trials are continuous.
b. each trial result influences the next.
c. the time between trials is constant.
d. each trial has exactly two outcomes whose probabilities do not change.
Chapter 3 – Probability Distributions
26. In a Poisson probability problem, the rate of errors is one every two hours. To find the probability of three defects in
four hours,
a. λ = 1, x = 4
b. λ = 2, x = 3
c. λ = 3, x = 2
d. λ = 3, x = 6
27. A probability density function
a. gives the probability that the random variable equals a specific value of x.
b. is used for discrete random variables.
c. describes the graph whose underlying area represents probability over an interval.
d. each of these choices are true.
28. The uniform distribution defined over the interval from 25 to 40 has the probability density function
a. f(x) = 1/40 for all x
b. f(x) = 5/8 for 25 ≤ x ≤ 40 and f(x) = 0 elsewhere
c. f(x) = 1/25 for 0 ≤ x ≤ 25 and f(x) = 1/40 for 26 ≤ x ≤ 40
d. f(x) = 1/15 for 25 ≤ x ≤ 40 and f(x) = 0 elsewhere
29. If x is normally distributed with mean 12 and standard deviation 2, then P(x ≤ 9) is
a. P(z ≤ 9/10).
b. P(z ≤ −3/2)
c. P(z ≤ 2/3)
30. When the exponential distribution probability is given as P(x ≤ 9 ) = 1 − e−9/18 the average value for x is
a. 18
b. 9
c. 9/18
d. 1/2
Chapter 3 – Probability Distributions
31. An experiment consists of measuring the speed of automobiles on a highway by the use of radar equipment. The
random variable in this experiment is speed, measured in miles per hour. This random variable is a
a. discrete random variable
b. continuous random variable
c. complex random variable
d. None of these alternatives is correct.
32. The weight of an object, measured to the nearest gram, is an example of
a. a continuous random variable
b. a discrete random variable
c. either a continuous or a discrete random variable, depending on the weight of the object
d. either a continuous or a discrete random variable depending on the units of measurement
33. Larger values of the standard deviation result in a normal curve that is
a. shifted to the right
b. shifted to the left
c. narrower and more peaked
d. wider and flatter
34. A standard normal distribution is a normal distribution with
a. a mean of 1 and a standard deviation of 0
b. a mean of 0 and a standard deviation of 1
c. any mean and a standard deviation of 1
d. any mean and any standard deviation
35. z is a standard normal random variable. The P(z ≥ 2.11) equals
a. 0.4821
b. 0.9821
c. 0.5
d. 0.0174
36. A probability distribution showing the probability of x successes in n trials, where the probability of success does not
change from trial to trial, is termed a
Chapter 3 – Probability Distributions
a. uniform probability distribution
b. binomial probability distribution
c. Poisson probability distribution
d. normal probability distribution
37. The expected value for a binomial probability distribution is
a. E(x) = np
b. E(x) = p(1 − p)
c. E(x) = np(1 − p)
d. E(x) = np(1 − n)
38. The Poisson probability distribution is used with
a. a discrete random variable
b. a continuous random variable
c. either a discrete or continuous random variable
d. any random variable
39. When dealing with the number of occurrences of an event over a specified interval of time or space and when the
occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval, the
appropriate probability distribution is a
a. binomial distribution
b. Poisson distribution
c. normal distribution
d. hypergeometric distribution
40. The binomial probability distribution is most symmetric when
a. n is 30 or greater
b. n equals p
c. p approaches 1
d. p equals 0.5
41. Whenever the probability is proportional to the length of the interval, the random variable is
a. normally distributed
b. binomially distributed
Chapter 3 – Probability Distributions
c. uniformly distributed
d. exponentially distributed
42. Variance is
a. a measure of the average, or central value of a random variable
b. a measure of the dispersion of a random variable
c. the square root of the standard deviation
d. the sum of the deviation of data elements from the mean
43. Which of the following is not a property of a binomial experiment?
a. the experiment consists of a sequence of n identical trials
b. each outcome can be referred to as a success or a failure
c. the probabilities of the two outcomes can change from one trial to the next
d. the trials are independent
44. If you are conducting an experiment where the probability of a success is .02 and you are interested in the probability
of 4 successes in 15 trials, the correct probability function to use is the
a. standard normal probability density function
b. normal probability density function
c. Poisson probability function
d. binomial probability function
45. If one wanted to find the probability of ten customer arrivals in an hour at a service station, one would generally use
the
a. binomial probability distribution
b. Poisson probability distribution
c. hypergeometric probability distribution
d. exponential probability distribution
46. Experimental outcomes that are based on measurement scales such as time, weight, and distance can be described by
_____ random variables.
a. discrete
b. continuous
c. uniform
d. intermittent
Chapter 3 – Probability Distributions
47. A binomial probability distribution with p = .3 is
a. negatively skewed
b. symmetrical
c. positively skewed
d. bimodal
48. Variance is
a. a measure of the average, or central value of a random variable
b. a measure of the dispersion of a random variable
c. the square root of the standard deviation
d. the sum of the deviation of data elements from the mean
49. Which of the following is not a property of a binomial experiment?
a. the experiment consists of a sequence of n identical trials
b. each outcome can be referred to as a success or a failure
c. the probabilities of the two outcomes can change from one trial to the next
d. the trials are independent
50. If you are conducting an experiment where the probability of a success is .02 and you are interested in the probability
of 4 successes in 15 trials, the correct probability function to use is the
a. standard normal probability density function
b. normal probability density function
c. Poisson probability function
d. binomial probability function
51. If one wanted to find the probability of ten customer arrivals in an hour at a service station, one would generally use
the
a. binomial probability distribution
b. Poisson probability distribution
c. uniform probability distribution
d. exponential probability distribution
52. Experimental outcomes that are based on measurement scales such as time, weight, and distance can be described by
_____ random variables.
a. discrete
b. continuous
c. uniform
d. intermittent
Chapter 3 – Probability Distributions
53. A binomial probability distribution with p = .3 is
a. negatively skewed
b. symmetric
c. positively skewed
d. bimodal
54. Z is a standard normal random variable. The P(1.20 z 1.85) equals
a. 0.4678
b. 0.3849
c. 0.8527
d. 0.0829
55. Z is a standard normal random variable. The P(1.05 z 2.13) equals
a. 0.8365
b. 0.1303
c. 0.4834
d. 0.6618
56. Z is a standard normal random variable. What is the value of z if the area to the right of z is 0.9803?
a. -2.06
b. 0.4803
c. 0.0997
d. 3.06
57. For a standard normal distribution, the probability of obtaining a z value of less than 1.6 is
a. 0.1600
b. 0.0160
c. 0.0016
Subjective Short Answer
58. Delicious Candy markets a two pound box of assorted chocolates. Because of imperfections in the candy making
equipment, the actual weight of the chocolate has a continuous uniform distribution ranging from 31.8 to 32.6 ounces.
a. Define a probability density function for the weight of the box of chocolate.
Chapter 3 – Probability Distributions
b. What is the probability that a box weighs (1) exactly 32 ounces; (2) more than 32.3 ounces; (3) less than 31.8
ounces?
c. The government requires that at least 60% of all products sold weigh at least as much as the stated weight. Is
Delicious violating government regulations?
59. During lunch time, customers arrive at Bob’s Drugs according to a Poisson distribution with λ = 4 per minute.
a. During a one-minute interval, determine the following probabilities: (1) no arrivals; (2) one arrival; (3) two arrivals;
and, (4) three or more arrivals.
b. What is the probability of two arrivals in a two-minute period?
c. What is the probability that no more than 30 seconds elapses between arrivals?
60. The number of customers at Winkies Donuts between 8:00a.m. and 9:00a.m. is believed to follow a Poisson
distribution with a mean of 2 customers per minute.
a. During a randomly selected one minute interval during this time period, what is the probability of 6 customers
arriving to Winkies?
b. What is the probability that at least 2 minutes elapse between customer arrivals?
61. The Harbour Island Ferry leaves on the hour and at 15 minute intervals. The time, x, it takes John to drive from his
house to the ferry has a uniform distribution with x between 10 and 20 minutes. One morning John leaves his house at
precisely 8:00a.m.
a. What is the probability John will wait less than 5 minutes for the ferry?
b. What is the probability John will wait less than 10 minutes for the ferry?
c. What is the probability John will wait less than 15 minutes for the ferry?
d. What is the probability John will not have to wait for the ferry?
e. Suppose John leaves at 8:05a.m. What is the probability John will wait (1) less than 5 minutes for the ferry; (2) less
than 10 minutes for the ferry?
f. Suppose John leaves at 8:10a.m. What is the probability John will wait (1) less than 5 minutes for the ferry; (2) less
than 10 minutes for the ferry?
g. What appears to be the best time for John to leave home if he wishes to maximize the probability of waiting less than
10 minutes for the ferry?
Chapter 3 – Probability Distributions
62. A light bulb manufacturer claims his light bulbs will last 500 hours on the average. The lifetime of a light bulb is
assumed to follow an exponential distribution.
a. What is the probability that the light bulb will have to replaced within 500 hours?
b. What is the probability that the light bulb will last more than 1000 hours?
c. What is the probability that the light bulb will last between 200 and 800 hours?
63. The high school GPAs of applicants for admission to a college program are recorded and relative frequencies are
calculated for the categories.
GPA f(x)
x < 2.0 .08
2.0 ≤ x < 2.5 .12
2.5 ≤ x < 3.0 .35
3.0 ≤ x < 3.5 .30
3.5 ≤ x
a. Complete the table to make this a valid probability distribution.
b. What is the probability an applicant’s GPA will be below 3.0?
c. What is the probability an applicant’s GPA will be 2.5 or above?
64. A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as
needed until a passing grade is obtained. Following is a record of the number of students in a class of 20 who took the test
each number of times.
Students Number of Tests
10 1
7 2
Chapter 3 – Probability Distributions
2 3
1 4
a. Use the relative frequency approach to construct a probability distribution and show that it satisfies the required
condition.
b. Find the expected value of the number of tests taken.
c. Compute the variance.
d. Compute the standard deviation.
65. A video rental store has two video cameras available for customers to rent. Historically, demand for cameras has
followed this distribution. The revenue per rental is $40. If a customer wants a camera and none is available, the store
gives a $15 coupon for snacks.
Demand Relative Frequency Revenue Cost
0 .35 0 0
1 .30 40 0
2 .20 80 0
3 .10 80 15
4 .05 80 30
a. What is the expected demand for camera rentals?
b. What is the expected revenue from camera rentals?
c. What is the expected cost associated with camera rentals?
d. What is the expected profit from camera rentals?
66. A process follows the binomial distribution with n = 8 and p = .3. Find
a. P(x = 3)
b. P(x > 6)
c. P(x ≤ 2)
Chapter 3 – Probability Distributions
67. A manufacturer of computer disks has a historical defective rate of .001. What is the probability that in a batch of
1000 disks, 2 would be defective?
68. For a binomial distribution, compare P(x = 3) when n = 10 and p = .4 to P(x = 7) when n = 10 and p = .6.
69. After a severe winter, potholes develop in a state highway at the rate of 5.2 per mile. Thirty-five miles of this highway
pass through Washington County.
a. How many potholes would you expect to see in the county?
b. What is the probability of finding 8 potholes in 1 mile of highway?
70. Consider a Poisson probability distribution in a manufacturing process with an average of 3 flaws every 100 feet. Find
the probability of
a. no flaws in 100 feet
b. 2 flaws in 100 feet
c. 1 flaws in 150 feet
d. 3 or 4 flaws in 150 feet
71. The weight of a .5 cubic yard bag of landscape mulch in uniformly distributed over the interval from 38.5 to 41.5
pounds.
a. Give a mathematical expression for the probability density function.
b. What is the probability that a bag will weigh more than 40 pounds?
Chapter 3 – Probability Distributions
c. What is the probability that a bag will weigh less than 39 pounds?
d. What is the probability that a bag will weigh between 39 and 40 pounds?
72. The time it takes to travel from home to the office is normally distributed with μ = 25 minutes and σ = 5 minutes.
a. What is the probability the trip takes more than 20 minutes?
b. What is the probability the trip takes less than 15 minutes.
c. What is the probability the trip takes between 30 and 35 minutes?
d. What is the probability the trip takes more than 40 minutes?
73. Scores on an endurance test for cardiac patients are normally distributed with μ = 182 and σ = 24.
a. What is the probability a patient will score above 190?
b. What percentage of patients score below 170?
74. Customers at a popular restaurant that refuses reservations arrive according to the Poisson distribution at a rate of 4
parties every 5 minutes. What is the probability that there will be more than 2 minutes between arriving parties?
75. Assume that the time required to download a file from the Internet is exponentially distributed with mean equal to 4
minutes. What is the probability that a download will require at least 2 but not more than 4 minutes?
Chapter 3 – Probability Distributions
76. Sandy’s Pet Center grooms large and small dogs. It takes Sandy 40 minutes to groom a small dog and 70 minutes to
groom a large dog. Large dogs account for 20% of Sandy’s business. Sandy has 5 appointments tomorrow.
a. What is the probability that all 5 appointments tomorrow are for small dogs?
b. What is the probability that two of the appointments tomorrow are for large dogs?
c. What is the expected amount of time to finish all five dogs tomorrow?
77. Water’s Edge is a clothing retailer that promotes its products via catalog and accepts customer orders by all of the
conventional ways including the Internet. The company has gained a competitive advantage by collecting data about its
operations and the customer each time an order is processed.
Among the data collected with each order are: number of items ordered, total shipping weight of the order, whether or not
all items ordered were available in inventory, time taken to process the order, customer’s number of prior orders in the last
12 months, and method of payment.
For each of the six aforementioned variables, identify which of the variables are discrete and which are continuous.
78. Telephone calls arrive at the Global Airline reservation office in Louisville according to a Poisson distribution with a
mean of 1.2 calls per minute.
a. What is the probability of receiving exactly one call during a one-minute interval?
b. What is the probability of receiving at most 2 calls during a one-minute interval?
c. What is the probability of receiving at least two calls during a one-minute interval?
d. What is the probability of receiving exactly 4 calls during a five-minute interval?
e. What is the probability that at most 2 minutes elapse between one call and the next?
79. Before dawn Josh hurriedly packed some clothes for a job-interview trip while his roommate was still sleeping. He
reached in his disorganized sock drawer where there were five black socks and five navy blue socks, although they
appeared to be the same color in the dimly lighted room. Josh grabbed six socks, hoping that at least two, and preferably
four, of them were black to match the gray suit he had packed. With no time to spare, he then raced to the airport to catch
his plane.
a. What is the probability that Josh packed at least two black socks so that he will be dressed appropriately the day of his
interview?
b. What is the probability that Josh packed at least four black socks so that he will be dressed appropriately the latter day
of his trip as well?
Chapter 3 – Probability Distributions
80. The salespeople at Gold Key Realty sell up to 9 houses per month. The probability distribution of a salesperson selling
x houses in a month is as follows:
Sales (x) 0 1 2 3 4 5 6 7 8 9
Probability f (x) .05 .10 .15 .20 .15 .10 .10 .05 .05 .05
a. What are the mean and standard deviation for the number of houses sold by a salesperson per month?
b. Any salesperson selling more houses than the amount equal to the mean plus two standard deviations receives a
bonus. How many houses per month must a salesperson sell to receive a bonus?
81. Twenty-five percent of all resumes received by a corporation for a management position are from females. Fifteen
resumes will be received tomorrow.
a. Define the random variable in words for this experiment.
b. What is the probability that exactly 5 of the resumes will be from females?
c. What is the probability that fewer than 3 of the resumes will be from females?
d. What is the expected number of resumes from women?
e. What is the variance of the number of resumes from women?
82. A salesperson contacts eight potential customers per day. From past experience, we know that the probability of a
potential customer making a purchase is 0.10.
a. Define the random variable in words for this experiment.
b. What is the probability the salesperson will make exactly two sales in a day?
c. What is the probability the salesperson will make at least two sales in a day?
d. What percentage of days will the salesperson not make a sale?
e. What is the expected number of sales per day?
Chapter 3 – Probability Distributions
84. The time between arrivals of customers at the drive-up window of a bank follows an exponential probability
distribution with a mean of 10 minutes.
a. What is the probability that the arrival time between customers will be 7 minutes or less?
b. What is the probability that the arrival time between customers will be between 3 and 7 minutes?
85. The time required to assemble a part of a machine follows an exponential probability distribution with a mean of 14
minutes.
a. What is the probability that the part can be assembled in 7 minutes or less?
b. What is the probability that the part can be assembled between 3.5 and 7 minutes?
86. A local bank has determined that the daily balances of the checking accounts of its customers are normally distributed
with an average of $280 and a standard deviation of $20.
a. What percentage of its customers has daily balances of more than $275?
b. What percentage of its customers has daily balances less than $243?
c. What percentage of its customers’ balances is between $241 and $301.60?
Chapter 3 – Probability Distributions
87. The weekly earnings of fast food restaurant employees are normally distributed with a mean of $395. If only 1.1% of
the employees have a weekly income of more than $429.35, what is the value of the standard deviation of the weekly
earnings of the employees?
88. The monthly earnings of computer programmers are normally distributed with a mean of $4,000. If only 1.7 percent of
programmers have monthly incomes of less than $2,834, what is the value of the standard deviation of the monthly
earnings of the computer programmers?
Essay
89. One way a discrete probability distribution can be portrayed is in a table that lists values for x and f(x). Describe two
other ways distributions can be represented.
90. Explain how the binomial distribution could be used in quality control to sample 50 parts in a lot of 500.
91. Use a graph to illustrate, for a continuous random variable, the relationship among P(x = a), P(a < x < b), and P(a ≤ x
≤ b).
92. Explain the transformation from any normal distribution to the standard normal.
93. A uniform distribution is defined on the interval from 1 ≤ x ≤ 5. Graph it, and explain how to determine the density
function.
94. Explain how the Poisson and exponential probability distributions relate to each other and how they relate to waiting
line models.
Chapter 3 – Probability Distributions