Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
Chapter 5 - Discrete Probability Distributions
91. 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
92. The _____ probability function is based in part on the counting rule for combinations.
a.
binomial
b.
Poisson
c.
hypergeometric
d.
exponential
93. To compute the probability that in a random sample of n elements, selected without replacement, we will obtain x
successes, we would use the
a.
binomial probability distribution
b.
Poisson probability distribution
c.
hypergeometric probability distribution
d.
exponential probability distribution
94. 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
95. Which of the following properties of a binomial experiment is called the stationarity
a.
The experiment consists of n identical trials
b.
Two outcomes are possible on each trial
c.
The probability of success is the same for each trial
d.
The trials are independent
96. The function used to compute the probability of x successes in n trials, when the trials are dependent, is the
a.
binomial probability function
b.
Poisson probability function
c.
hypergeometric probability function
d.
exponential probability function
97. The expected value of a random variable is the
a.
most probable value
b.
simple average of all the possible values
c.
median value
d.
mean value
98. In a binomial experiment consisting of five trials, the number of different values that x (the number of successes) can
assume is
a.
2
b.
5
c.
6
d.
10
99. A binomial probability distribution with p = .3 is
a.
negatively skewed
b.
symmetrical
c.
positively skewed
d.
bimodal
100. In a Poisson probability problem, the rate of defects is one every two hours. To find the probability of three defects
in four hours,
a.
= 1, x = 4
b.
= 2, x = 3
c.
= 3, x = 4
d.
= 4, x = 3
101. 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
Chapter 5 - Discrete Probability Distributions
c.
the time between trials is constant
d.
each trial has exactly two outcomes whose probabilities do not change
102. In order to compute a binomial probability we must know all of the following except
a.
the probability of success
b.
the number of elements in the population
c.
the number of trials
d.
the value of the random variable
103. A property of the Poisson distribution is that the mean equals the
a.
mode
b.
median
c.
variance
d.
standard deviation
104. A discrete probability distribution for which the relative frequency method is used to assign probabilities is the
a.
binomial distribution
b.
empirical discrete distribution
c.
hypergeometric distribution
d.
discrete uniform distribution
105. A name closely associated with the binomial probability distribution is
a.
Bernoulli
b.
de Moivre
c.
Pareto
d.
Poisson
106. The probability distribution for the rate of return on an investment is
Rate of Return (%)
Probability
9.5
.1
9.8
.2
10.0
.3
10.2
.3
10.6
.1
a.
What is the probability that the rate of return will be at least 10%?
b.
What is the expected rate of return?
c.
What is the variance of the rate of return?
107. A random variable x has the following probability distribution:
x
f(x)
0
0.08
1
0.17
2
0.45
3
0.25
4
0.05
a.
Determine the expected value of x.
b.
Determine the variance.
108. For the following probability distribution:
x
f(x)
0
0.01
1
0.02
2
0.10
3
0.35
4
0.20
5
0.11
6
0.08
7
0.05
8
0.04
9
0.03
10
0.01
a.
Determine E(x).
b.
Determine the variance and the standard deviation.
109. A company sells its products to wholesalers in batches of 1,000 units only. The daily demand for its product and the
respective probabilities are given below.
Demand (Units)
Probability
0
0.2
1000
0.2
2000
0.3
3000
0.2
4000
0.1
a.
Determine the expected daily demand.
b.
Assume that the company sells its product at $3.75 per unit. What is the expected daily
revenue?
110. The demand for a product varies from month to month. Based on the past year's data, the following probability
distribution shows MNM company's monthly demand.
x
f(x)
Unit Demand
Probability
0
0.10
1,000
0.10
2,000
0.30
3,000
0.40
4,000
0.10
a.
Determine the expected number of units demanded per month.
b.
Each unit produced costs the company $8.00, and is sold for $10.00. How much will the
company gain or lose in a month if they stock the expected number of units demanded, but
sell 2000 units?
111. The probability distribution of the daily demand for a product is shown below.
Demand
Probability
0
0.05
1
0.10
2
0.15
3
0.35
4
0.20
5
0.10
6
0.05
a.
What is the expected number of units demanded per day?
b.
Determine the variance and the standard deviation.
112. The random variable x has the following probability distribution:
x
f(x)
0
.25
1
.20
2
.15
3
.30
Chapter 5 - Discrete Probability Distributions
4
.10
a.
Is this probability distribution valid? Explain and list the requirements for a valid probability
distribution.
b.
Calculate the expected value of x.
c.
Calculate the variance of x.
d.
Calculate the standard deviation of x.
113. The probability function for the number of insurance policies John will sell to a customer is given by
f(x) = .5 − (x/6) for x = 0, 1, or 2
a.
Is this a valid probability function? Explain your answer.
b.
What is the probability that John will sell exactly 2 policies to a customer?
c.
What is the probability that John will sell at least 2 policies to a customer?
d.
What is the expected number of policies John will sell?
e.
What is the variance of the number of policies John will sell?
114. Thirty-two percent of the students in a management class are graduate students. A random sample of 5 students is
selected. Using the binomial probability function, determine the probability that the sample contains exactly 2 graduate
students?
115. A production process produces 2% defective parts. A sample of 5 parts from the production is selected. What is the
probability that the sample contains exactly two defective parts? Use the binomial probability function and show your
computations to answer this question.
116. When a particular machine is functioning properly, 80% of the items produced are non-defective. If three items are
examined, what is the probability that one is defective? Use the binomial probability function to answer this question.
117. The records of a department store show that 20% of its customers who make a purchase return the merchandise in
order to exchange it. In the next six purchases,
a.
what is the probability that three customers will return the merchandise for exchange?
b.
what is the probability that four customers will return the merchandise for exchange?
c.
what is the probability that none of the customers will return the merchandise for exchange?
118. Ten percent of the items produced by a machine are defective. Out of 15 items chosen at random,
a.
what is the probability that exactly 3 items will be defective?
b.
what is the probability that less than 3 items will be defective?
c.
what is the probability that exactly 11 items will be non-defective?
119. In a large university, 15% of the students are female. If a random sample of twenty students is selected,
a.
what is the probability that the sample contains exactly four female students?
b.
what is the probability that the sample will contain no female students?
c.
what is the probability that the sample will contain exactly twenty female students?
d.
what is the probability that the sample will contain more than nine female students?
e.
what is the probability that the sample will contain fewer than five female students?
f.
what is the expected number of female students?
120. Seventy percent of the students applying to a university are accepted. What is the probability that among the next 18
applicants
a.
At least 6 will be accepted?
b.
Exactly 10 will be accepted?
c.
Exactly 5 will be rejected?
d.
Fifteen or more will be accepted?
e.
Determine the expected number of acceptances.
f.
Compute the standard deviation.
f.
1.9442
POINTS:
1
121. Twenty percent of the applications received for a particular position are rejected. What is the probability that among
the next fourteen applications,
a.
none will be rejected?
b.
all will be rejected?
c.
less than 2 will be rejected?
d.
more than four will be rejected?
e.
Determine the expected number of rejected applications and its variance.
Chapter 5 - Discrete Probability Distributions
122. Fifty-five percent of the applications received for a particular credit card are accepted. Among the next twelve
applications,
a.
what is the probability that all will be rejected?
b.
what is the probability that all will be accepted?
c.
what is the probability that exactly 4 will be accepted?
d.
what is the probability that fewer than 3 will be accepted?
e.
Determine the expected number and the variance of the accepted applications.
123. In a southern state, it was revealed that 5% of all automobiles in the state did not pass inspection. Of the next ten
automobiles entering the inspection station,
a.
what is the probability that none will pass inspection?
b.
what is the probability that all will pass inspection?
c.
what is the probability that exactly two will not pass inspection?
d.
what is the probability that more than three will not pass inspection?
e.
what is the probability that fewer than two will not pass inspection?
f.
Find the expected number of automobiles not passing inspection.
g.
Determine the standard deviation for the number of cars not passing inspection.
124. Only 0.02% of credit card holders of a company report the loss or theft of their credit cards each month. The
company has 15,000 credit cards in the city of Memphis. What is the probability that during the next month in the city of
Memphis
a.
no one reports the loss or theft of their credit cards?
b.
every credit card is lost or stolen?
c.
six people report the loss or theft of their cards?
d.
at least nine people report the loss or theft of their cards?
e.
Determine the expected number of reported lost or stolen credit cards.
f.
Determine the standard deviation for the number of reported lost or stolen cards.
Chapter 5 - Discrete Probability Distributions
125. Two percent of the parts produced by a machine are defective. Forty parts are selected. Define the random variable x
to be the number of defective parts.
a.
What is the probability that exactly 3 parts will be defective?
b.
What is the probability that the number of defective parts will be more than 2 but fewer than 6?
c.
What is the probability that fewer than 4 parts will be defective?
d.
What is the expected number of defective parts?
e.
What is the variance for the number of defective parts?
126. A manufacturing company has 5 identical machines that produce nails. The probability that a machine will break
down on any given day is 0.1. Define a random variable x to be the number of machines that will break down in a day.
a.
What is the appropriate probability distribution for x? Explain how x satisfies the properties of
the distribution.
b.
Compute the probability that 4 machines will break down.
c.
Compute the probability that at least 4 machines will break down.
d.
What is the expected number of machines that will break down in a day?
e.
What is the variance of the number of machines that will break down in a day?
127. In a large corporation, 65% of the employees are male. A random sample of five employees is selected.
a.
Define the random variable in words for this experiment.
b.
What is the probability that the sample contains exactly three male employees?
c.
What is the probability that the sample contains no male employees?
d.
What is the probability that the sample contains more than three female employees?
e.
What is the expected number of female employees in the sample?
128. In a large university, 75% of students live in the dormitories. A random sample of 5 students is selected.
a.
Define the random variable in words for this experiment.
b.
What is the probability that the sample contains exactly three students who live in the
dormitories?
Chapter 5 - Discrete Probability Distributions
c.
What is the probability that the sample contains no students who live in the dormitories?
d.
What is the probability that the sample contains more than three students who do not live in the
dormitories?
e.
What is the expected number of students (in the sample) who do not live in the dormitories?
129. A production process produces 90% non-defective parts. A sample of 10 parts from the production process is
selected.
a.
Define the random variable in words for this experiment.
b.
What is the probability that the sample will contain 7 non-defective parts?
c.
What is the probability that the sample will contain at least 4 defective parts?
d.
What is the probability that the sample will contain less than 5 non-defective parts?
e.
What is the probability that the sample will contain no defective parts?
130. The student body of a large university consists of 30% Business majors. A random sample of 20 students is selected.
a.
Define the random variable in words for this experiment.
b.
What is the probability that among the students in the sample at least 10 are Business majors?
c.
What is the probability that at least 16 are not Business majors?
d.
What is the probability that exactly 10 are Business majors?
e.
What is the probability that exactly 12 are not Business majors?
131. A local university reports that 3% of their students take their general education courses on a pass/fail basis. Assume
that fifty students are registered for a general education course.
a.
Define the random variable in words for this experiment.
b.
What is the expected number of students who have registered on a pass/fail basis?
c.
What is the probability that exactly five are registered on a pass/fail basis?
d.
What is the probability that more than three are registered on a pass/fail basis?
e.
What is the probability that less than four are registered on a pass/fail basis?
132. Twenty-five percent of the employees of a large company are minorities. A random sample of 7 employees is
selected.
a.
Define the random variable in words for this experiment.
b.
What is the probability that the sample contains exactly 4 minorities?
c.
What is the probability that the sample contains fewer than 2 minorities?
d.
What is the probability that the sample contains exactly 1 non-minority?
e.
What is the expected number of minorities in the sample?
f.
What is the variance of the minorities?
133. 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?
134. 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?
135. An insurance company has determined that each week an average of nine claims are filed in their Atlanta branch.
What is the probability that during the next week
a.
exactly seven claims will be filed?
b.
no claims will be filed?
c.
less than four claims will be filed?
d.
at least eighteen claims will be filed?
Chapter 5 - Discrete Probability Distributions
136. John parks cars at a hotel. On the average, 6.7 cars will arrive in an hour. Assume that a driver's decision on whether
to let John park the car does not depend upon any other person's decision. Define the random variable x to be the number
of cars arriving in any hour period.
a.
What is the appropriate probability distribution for x? Explain how x satisfies the properties of
the distribution.
b.
Compute the probability that exactly 5 cars will arrive in the next hour.
c.
Compute the probability that no more than 5 cars will arrive in the next hour.
137. The average number of calls received by a switchboard in a 30-minute period is 15.
a.
Define the random variable in words for this experiment.
b.
What is the probability that between 10:00 and 10:30 the switchboard will receive exactly 10
calls?
c.
What is the probability that between 10:00 and 10:30 the switchboard will receive more than 9
calls but fewer than 15 calls?
d.
What is the probability that between 10:00 and 10:30 the switchboard will receive fewer than 7
calls?
138. A life insurance company has determined that each week an average of seven claims is filed in its Nashville branch.
a.
Define the random variable in words for this experiment.
b.
What is the probability that during the next week exactly seven claims will be filed?
c.
What is the probability that during the next week no claims will be filed?
d.
What is the probability that during the next week fewer than four claims will be filed?
e.
What is the probability that during the next week at least seventeen claims will be filed?
139. General Hospital has noted that they admit an average of 8 patients per hour.
a.
Define the random variable in words for this experiment.
b.
What is the probability that during the next hour fewer then 3 patients will be admitted?
c.
What is the probability that during the next two hours exactly 8 patients will be admitted?
140. Shoppers enter Hamilton Place Mall at an average of 120 per hour.
a.
Define the random variable in words for this experiment.
b.
What is the probability that exactly 5 shoppers will enter the mall between noon and 1:00 p.m.?
c.
What is the probability that exactly 5 shoppers will enter the mall between noon and 12:05
p.m.?
d.
What is the probability that at least 35 shoppers will enter the mall between 5:00 and 5:10
p.m.?
141. Compute the hypergeometric probabilities for the following values of n and x. Assume N = 8 and r = 5.
a.
n = 5, x = 2
b.
n = 6, x = 4
c.
n = 3, x = 0
d.
n = 3, x = 3
142. A retailer of electronic equipment received six HDTVs from the manufacturer. Three of the HDTVs were damaged
in the shipment. The retailer sold two HDTVs to two customers.
a
Can a binomial formula be used for the solution of the above problem?
b.
What kind of probability distribution does the above satisfy, and is there a function for
solving such problems?
c.
What is the probability that both customers received damaged HDTVs?
d.
What is the probability that one of the two customers received a defective HDTV?
143. Waters’ 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.
144. June's Specialty Shop sells designer original dresses. On 10% of her dresses, June makes a profit of $10, on 20% of
her dresses she makes a profit of $20, on 30% of her dresses she makes a profit of $30, and on 40% of her dresses she
makes a profit of $40. On a given day, the probability of June having no customers is .05, of one customer is .10, of two
customers is .20, of three customers is .35, of four customers is .20, and of five customers is .10.
a. What is the expected profit June earns on the sale of a dress?
b. June's daily operating cost is $40 per day. Find the expected net profit June earns per day. (Hint: To find the expected
daily gross profit, multiply the expected profit per dress by the expected number of customers per day.)
c. June is considering moving to a larger store. She estimates that doing so will double the expected number of customers.
If the larger store will increase her operating costs to $100 per day, should she make the move?
145. 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?
146. 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?
147. Ralph's Gas Station is running a giveaway promotion. With every fill-up of gasoline, Ralph gives out a lottery ticket
that has a 25% chance of being a winning ticket. Customers who collect four winning lottery tickets are eligible for the
"BIG SPIN" for large payoffs. What is the probability of qualifying for the big spin if a customer fills up: (a) 3 times; (b)
4 times; (c) 7 times?
148. 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
Chapter 5 - Discrete Probability Distributions
to Winkies?
b. What is the probability that at least 2 minutes elapse between customer arrivals?
149. During lunchtime, 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?
150. 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?
151. 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?
152. Consider a Poisson probability distribution in a 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
153. 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?
154. A manufacturer of computer disk drives has a historical defective rate of .001. What is the probability that in a batch
of 1000 drives, 2 would be defective?
155. For a binomial distribution, compare P(x = 3) when n = 10 and p = .4, to P(x = 7) when n = 10 and p = .6.
156. A 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?
157. 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
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.
158. A pet ownership survey was conducted by Stanton Marketing for Dollar Department Store. Out of 400 families
surveyed, 260 owned no pet, 120 owned dogs and 50 owned cats.
a) On the basis of this information, find the probability distribution for the random variable x, where x = 1 if own no pet,
x = 2 if own dog(s) only, x = 3 if own cat(s) only, and x = 4 if own dog(s) and cat(s).
b) Dollar Department Stores is considering opening a pet department if the expected number of families owning pets
shopping at its store exceeds 4,000. If Dollar expects to serve 12,000 families, should it open a pet department?
159. 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 on August 15.
a) What is the probability that all 5 dogs are small?
b) What is the probability that two of the dogs are large?
c) What is the expected amount of time to finish all five dogs?
160. June's Specialty Shop sells designer original dresses. On 10% of her dresses, June makes a profit of $10, on 20% of
her dresses she makes a profit of $20, on 30 of her dresses she makes a profit of $30, and on 40% of her dresses she
makes a profit of $40.
a) What is the expected profit June earns on the sale of a dress?
b) On a given day, the probability of June having no customers is .05, one customer is .10, two customers is .20, three
customers is .35, four customers is .20, and five customers is .10. June's daily operating cost is $0 per day. Using your
answer to (a), find the expected net profit June earns per day. (Hint: To find the expected daily gross profit, multiply the
expected profit per dress by the expected number of customers per day.)
c) June is considering moving to a larger store. She estimates that doing so will double the expected number of
customers. If the larger store will increase her operating costs to $100 per day, should she make the move?
161. Chez Paul is an exclusive French restaurant that seats only 10 couples for dinner. Paul is famous for his "truffle
salad for two" which must be prepared one day in advance. The probability of any couple ordering the salad is .4 and
each couple orders independently of other couples.
a) What is the expected number of "truffle salads for two" that Paul serves per dinner? What is the variance?
b) What is the probability that on a given evening, at most three couples want a "truffle salad for two"?
c) How many salads should Paul prepare if he wants the probability of not having enough salads for all customers who
desire one to be no greater than .10?
d) There is a 70% chance a couple will order coffee after dinner. What is the probability that on a given evening, exactly
eight of the ten couples will order coffee? (Again assume that couples order independently of each other.)
