101. Bivariate probabilities are often called
a.
union probabilities
b.
conditional probabilities
c.
marginal probabilities
d.
joint probabilities
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
PROBLEM
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?
a.
b.
c.
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?
a.
2.02
b.
0.9396
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?
a.
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.
b.
c.
d.
e.
b.
c.
d.
e.
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.
b.
1.8
c.
1.86
d.
1.364
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.
b.
c.
a.
b.
c.
13. Ten percent of the items produced by a machine are defective. Out of 15 items chosen at random,
a.
b.
c.
a.
b.
c.
14. In a large university, 15% of the students are female. If a random sample of twenty students is
selected,
a.
b.
c.
d.
e.
f.
a.
c.
d.
e.
15. Seventy percent of the students applying to a university are accepted. What is the probability that
among the next 18 applicants
a.
b.
c.
d.
e.
f.
16. Twenty percent of the applications received for a particular position are rejected. What is the
probability that among the next fourteen applications,
a.
b.
c.
d.
e.
a.
b.
c.
d.
17. Fifty-five percent of the applications received for a particular credit card are accepted. Among the next
twelve applications,
a.
b.
c.
d.
e.
b.
c.
d.
e.
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.
a.
0.9988
b.
0.0811
c.
0.2017
d.
0.1646
e.
12.6
f.
1.9442
b.
c.
d.
e.
f.
g.
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.
b.
c.
d.
e.
f.
b.
c.
d.
e.
f.
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.
b.
c.
d.
e.
a.
b.
c.
d.
e.
a.
0.0000
b.
0.5987
c.
0.0746
d.
0.0010
e.
0.9139
0.5
g.
0.6892
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.
b.
c.
d.
e.
a.
b.
c.
e.
22. In a large corporation, 65% of the employees are male. A random sample of five employees is
selected.
a.
b.
c.
d.
e.
b.
c.
d.
e.
23. In a large university, 75% of students live in the dormitories. A random sample of 5 students is
selected.
a.
b.
c.
d.
e.
b.
c.
d.
24. A production process produces 90% non-defective parts. A sample of 10 parts from the production
process is selected.
a.
b.
c.
d.
e.
b.
c.
d.
e.
25. The student body of a large university consists of 30% Business majors. A random sample of 20
students is selected.
a.
b.
c.
d.
e.
b.
c.
d.
e.
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.
b.
c.
d.
e.
b.
c.
d.
e.
1.25
27. Twenty-five percent of the employees of a large company are minorities. A random sample of 7
employees is selected.
a.
b.
c.
d.
e.
f.
b.
c.
d.
f.
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.
b.
c.
d.
e.
b.
c.
d.
e.
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.
b.
c.
d.
e.
b.
c.
d.
30. An insurance company has determined that each week an average of nine claims is filed in their
Atlanta branch. What is the probability that during the next week
a.
b.
c.
d.
a.
b.
c.
d.
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.
b.
c.
a.
b.
c.
32. The average number of calls received by a switchboard in a 30-minute period is 15.
a.
b.
c.
d.
b.
c.
d.
33. A life insurance company has determined that each week an average of seven claims is filed in its
Nashville branch.
a.
b.
c.
d.
e.
34. General Hospital has noted that they admit an average of 8 patients per hour.
a.
b.
c.
b.
c.
35. Shoppers enter Hamilton Place Mall at an average of 120 per hour.
a.
b.
c.
d.
b.
c.
d.
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
a.
b.
c.
d.
b.
0.1490
c.
0.0009
d.
0.0818
e.
0.0010
37. A retailer of electronic equipment received six Blu-ray players from the manufacturer. Three of the
players were damaged in the shipment. The retailer sold two players 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 players?
d.
What is the probability that one of the two customers received a defective player?
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?
47. 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