MGNT 4640, PROBABILITY PROBLEMS
1.
Mary Ann Vance is an avid baseball fan. At the beginning of the season, she estimated that
the New York Mets had a .55 probability of winning the National League East and that the
Philadelphia Phillies had a .20 probability of winning the NL East. According to her
subjective estimates, what is the probability that either the Mets or the Phillies will win the
NL East?
2.
Based upon a computer analysis by sportswriters at the end of the regular season, it is
estimated that the San Antonio Spurs have a 30% chance to reach the NBA Finals and the
Detroit Pistons have a 25% chance. There is a 10% probability that both will reach the
Finals.
(a)
(b)
3.
What is the probability that either the Spurs or the Pistons will reach the Finals?
What is the probability that neither will reach the Finals?
The Iverson Manufacturing Company makes four types of screws: long fine-thread, long
coarse-thread, short fine-thread, and short coarse-thread. A box contains an assortment of
50 screws. There are 22 short screws, 15 screws that are long and coarse, and 38 that are
long or fine.
(a)
(b)
(c)
Construct a 2 by 2 table to determine the number of screws in each of the four
categories.
What is the probability that a screw drawn at random is either short or fine?
What is the probability that a screw drawn at random is either short or coarse?
4.
A die is tossed three times. What is the probability that the first toss will show an odd
number, the second toss will be even, and the third toss will be a five?
5.
David Young is analyzing his sales calls. Last month, out of 300 calls, he made 81 sales.
Sixty of the calls were in the Lake Park area; the rest were in Valdosta. Assume that calls
are independent of each other and that the likelihood of a sale is independent of location.
What is the probability that a given call will be: (a) A no-sale call? (b) A Lake Park call?
(c) A Valdosta call and a call that resulted in a sale?
6.
An automatic machine makes bolts and fills boxes with them. On the average, 1 box in 25
has at least one defective bolt in it. The quality within each box is independent of the
quality in other boxes. Three boxes are chosen at random. What is the probability that all
three of them contain at least one defective bolt?
7.
On homecoming weekend, four sports events will be held at Thomasville State College:
football, tennis, golf, and cross-country. Thomasville's coaches have estimated probabilities
of winning to be .80, .70, .60, and .40, respectively. Assume that the events are
independent.
(a)
(b)
(c)
What is the probability that the football and golf teams will win?
What is the probability that either the football or golf team will win?
What is the probability that all four teams will win?
PROBABILITY PROBLEMS, PAGE 2
8.
Three different items A, B, C are produced simultaneously and sent down a conveyor belt.
At a quality control check point, selected items are inspected and then accepted or rejected.
(The quality of Item A is independent of the quality of Items B and C.) Last week, the
record of acceptances and rejections were as shown below.
Rejected
Accepted
Item A
20
300
Item B
120
280
Item C
60
380
What is the probability that the three items (one of each) appearing on the conveyor at the
same time will all be rejected? Use 4-decimal place accuracy.
9.
In one department of a machine shop, 40% of the repair jobs are drilling and 60% are
welding.
(a)
(b)
(c)
10.
The Duncan Company sells two types of photocopiers, Model A and Model B, by sending
brochures about both types to prospective customers. Based upon past experience, 30% of
customers buy Model A, 20% buy Model B, and 10% buy both models. For each part
below, define appropriate events, state the given probabilities, and state the formula used.
(a)
(b)
11.
What is the probability that two jobs in a row will be drilling jobs?
What is the probability that three jobs in a row will be welding jobs?
What is the probability that exactly two of the next three jobs will be drilling jobs?
What is the probability that a customer will buy at least one of the two models?
Given that a customer has bought Model A, what is the probability that he will also
buy Model B?
The marketing vice-president of Emmitt's Appliances demonstrated a new kitchen appliance
at the annual dealers' meeting. After the demonstration, he took a sample of 200 dealers and
obtained the following data: 140 of the dealers were men and 60 were women. Forty of the
men said they wanted to stock the new product in their store; 25 of the women wanted to
stock the new product. For each question below, define appropriate events, state the given
probabilities, and state the formula used.
(a)
(b)
(c)
(d)
Given that a dealer is a woman, what is the probability that the new appliance will be
stocked?
Given that a dealer is a man, what is the probability that the new appliance will be
stocked?
Given that a dealer stocks an item, what is the probability that the dealer is a man?
Given that a dealer stocks an item, what is the probability that the dealer is a woman?
PROBABILITY PROBLEMS, PAGE 3
12.
After studying the relationship between interest rates and new car sales for the past few
years, Langdale Buick Company has estimated several probabilities. There is a .55
probability that interest rates will increase next year. There is a .25 probability of both an
increase in interest rates and an increase in new car sales next year. There is a .35 probability that interest rates will not increase and new car sales will increase. For each part below,
define appropriate events, state the given probabilities, and state the formula used.
(a)
(b)
If interest rates increase next year, what is the probability that new car sales will
increase?
If interest rates do not increase next year, what is the probability that new car sales will
increase?
13.
Among Valdosta adults, the probability that a person is a medical doctor is .005 . Given that
a person is a doctor, the probability that he/she has an annual income greater than $100,000
is .80. What is the probability that an adult in Valdosta is a doctor and has an income
greater than $100,000? Define appropriate events, state the given probabilities, and state the
formula used.
14.
Bret Gibson is dealt three cards. What is the probability that all three are hearts?
15.
Suppose you are considering buying some Wal-Mart stock. You estimate that if the Dow
Jones average increases next year, then there is an 80% probability that Wal-Mart stock will
go up also. You also believe that there is a 75% percent chance that the Dow Jones average
will increase next year. What is the probability that both the Dow Jones average and WalMart stock will go up next year? Define appropriate events, state the given probabilities,
and state the formula used.
16.
In a factory, Machine A produces 30% of the output, Machine B produces 25%, and
Machine C produces 45%. One percent of Machine A's output is defective, 1.2% of
Machine B's output is defective, and 2% of Machine C's output is defective.
(a)
(b)
(c)
17.
If a randomly drawn item is defective, what is the probability that is was produced by
Machine A?
If a randomly drawn item is defective, what is the probability that is was produced by
Machine B? By Machine C?
For Part (a), explain why the revised probability that an item was produced by
Machine A is less than the prior probability that an item was produced by Machine A.
An auto insurance company classifies drivers as Class A (good risks), Class B (medium
risks), and Class C (poor risks). They expect that 30%, 50%, and 20% of the applicants for
insurance are Class A, Class B, and Class C, respectively. The probability that a driver will
have an accident within a 12-month period is .01, .03, and .10, for the 3 classes.
(a)
(b)
If a policy buyer has an accident within 12 months, what is the probability that he/she
is a Class A risk? Class B risk?
If a policy buyer has an accident within 12 months, what is the probability that he/she
is a Class C risk?
PROBABILITY PROBLEMS, PAGE 4
17(c) For Part (b), explain why the revised probability of a Class C risk is greater than the prior
probability of a Class C risk.
18.
In the Brownwood, Texas school system, 30 percent of the students are from low-income
families. Students from low-income families graduate from high school only 50 percent of
the time; students that are not from low-income families have a 90 percent probability of
graduating. A job applicant at a department store just graduated from Brownwood High
School. What is the probability that the applicant was from a low-income family? Explain
carefully why the revised probability of a low-income family was less than the prior
probability.
19.
At Oklahoma Loan Company, ninety percent of applicants state their income truthfully on
their loan applications. If an applicant's application is truthful, the probability of the
applicant being a good credit risk is .8 . If an applicant's application is not truthful, the
probability of the applicant being a good credit risk is .3 . If an applicant turns out to be a
good credit risk, what is the probability that his application was truthful?
20.
Oil drilling companies can use seismic tests to aid in the decision concerning whether or not
to drill at a particular location. Among all wells drilled in the past by the Keefe Oil
Company, 60% had no oil, 30% had small amounts of oil, and 10% had large amounts of
oil. Of wells which had no oil, 20% had favorable seismic results. Of wells which small
amounts of oil, 50% had favorable seismic results. Of wells which had large amounts of oil,
80% had favorable seismic results. Suppose that a seismic test has just been done at a new
drilling site.
(a)
(b)
If the seismic results are favorable, what is the probability that the site contains large
amounts of oil? Define events, show formulas and calculations, and express the final
answer as a 2-place decimal.
What are the prior and revised probabilities of large amounts of oil? Explain how and
why these probabilities differ as they do.
ANSWERS TO PROBABILITY PROBLEMS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
.75
(a) .45
(b) .70
1/24
(a) .73
.000064
(a) .48
.0026
(a) .16
(a) .40
(b) .55
(c) .74
(b) .20 (c) .216
(b) .92 (c) .1344
(b) .216 (c) .288
(b) .33
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
(a) .42 (b) .29 (c) .62 (d) .38
(a) .45 (b) .78
.004
.0129
.60
(a) .20 (b) .20, .60
(a) .079, .395 (b) .526
.19
.96
.23