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c. 10
d. 2.30
93. Refer to Exhibit 5-11. The probability that there are 8 occurrences in ten minutes is
a. .0241
b. .0771
c. .1126
d. .9107
94. Refer to Exhibit 5-11. The probability that there are less than 3 occurrences is
a. .0659
b. .0948
c. .1016
d. .1239
95. When sampling without replacement, the probability of obtaining a certain sample is best
given by a
a. hypergeometric distribution
b. binomial distribution
c. Poisson distribution
d. normal distribution
96. The key difference between the binomial and hypergeometric distribution is that with the
hypergeometric distribution the
a. probability of success must be less than 0.5
b. probability of success changes from trial to trial
c. trials are independent of each other
d. random variable is continuous
97. Excel’s HYPGEOMDIST function can be used to compute
a. bin width for histograms
b. hypergeometric probabilities
c. cumulative hypergeometric probabilities
d. Both hypergeometric probabilities and cumulative hypergeometric probabilities
are correct.
98. Excel’s HYPGEOMDIST function has how many inputs?
a. 2
b. 3
c. 4
d. 5
99. When using Excel’s HYPGEOMDIST function, one should choose TRUE for the fourth
input if
a. a probability is desired
b. a cumulative probability is desired
c. the expected value is desired
d. None of the alternative answers is correct.
PROBLEMS
1. 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?
2. 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.
3. 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.
4. 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?
5. 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?
6. 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.
7. The random variable x has the following probability distribution:
x
f(x)
0
.25
1
.20
2
.15
3
.30
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.
8. 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?
9. 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?
10. 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.
11. 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.
12. 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?
13. 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?
14. 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?
15. 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.
16. 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.
17. 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.
18. 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.
19. 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.
20. 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?
21. 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?
22. 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?
23. 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?
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?
24. 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?
25. 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?
26. 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?
27. 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?
28. 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?
29. 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?
30. 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?
31. 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.
32. 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?
33. 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?
34. 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?
35. 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.?
36. 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
37. A retailer of electronic equipment received six VCRs from the manufacturer. Three of
the VCRs were damaged in the shipment. The retailer sold two VCRs 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 VCRs?
d. What is the probability that one of the two customers received a defective VCR?
38. 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.
39. 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?
40. 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?
41. 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?
42. 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?
43. 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?
44. During lunchtime, customers arrive at Bob's Drugs according to a Poisson distribution
with = 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?
45. 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?
46. 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?
