Chapter 2—Introduction to Probability

PROBLEM

1. A market study taken at a local sporting goods store showed that of 20 people questioned, 6 owned tents, 10 owned sleeping bags, 8 owned camping stoves, 4 owned both tents and camping stoves, and 4 owned both sleeping bags and camping stoves.

Let: / Event A = owns a tent
Event B = owns a sleeping bag
Event C = owns a camping stove

and let the sample space be the 20 people questioned.

a. / Find P(A), P(B), P(C), P(A C), P(B C).
b. / Are the events A and C mutually exclusive? Explain briefly.
c. / Are the events B and C independent events? Explain briefly.
d. / If a person questioned owns a tent, what is the probability he also owns a camping stove?
e. / If two people questioned own a tent, a sleeping bag, and a camping stove, how many own only a camping stove? In this case is it possible for 3 people to own both a tent and a sleeping bag, but not a camping stove?

ANSWER:

a. / P(A) = .3; P(B) = .5; P(C) = .4; P(A B) = .2; P(B C) = .2
b. / Events B and C are not mutually exclusive because there are people (4 people) who both own a tent and a camping stove.
c. / Since P(B C) = .2 and P(B)P(C) = (.5)(.4) = .2, then these events are independent.
d. / .667
e. / Two people own only a camping stove; no, it is not possible

2. An accounting firm has noticed that of the companies it audits, 85% show no inventory shortages, 10% show small inventory shortages and 5% show large inventory shortages. The firm has devised a new accounting test for which it believes the following probabilities hold:

P(company will pass test | no shortage) / = .90
P(company will pass test | small shortage) / = .50
P(company will pass test | large shortage) / = .20
a. / If a company being audited fails this test, what is the probability of a large or small inventory shortage?
b. / If a company being audited passes this test, what is the probability of no inventory shortage?

ANSWER:

a. / .515
b. / .927

3. An investment advisor recommends the purchase of stock shares in Infomatics, Inc. He has made the following predictions:

P(Stock goes up 20% | Rise in GDP) / = .6
P(Stock goes up 20% | Level GDP) / = .5
P(Stock goes up 20% | Fall in GDP) / = .4

An economist has predicted that the probability of a rise in the GDP is 30%, whereas the probability of a fall in the GDP is 40%.

a. / What is the probability that the stock will go up 20%?
b. / We have been informed that the stock has gone up 20%. What is the probability of a rise or fall in the GDP?

ANSWER:

a. / .49
b. / .367 + .327 = .694

4. Global Airlines operates two types of jet planes: jumbo and ordinary. On jumbo jets, 25% of the passengers are on business while on ordinary jets 30% of the passengers are on business. Of Global's air fleet, 40% of its capacity is provided on jumbo jets. (Hint: The 25% and 30% values are conditional probabilities stated as percentages.)

a. / What is the probability a randomly chosen business customer flying with Global is on a jumbo jet?
b. / What is the probability a randomly chosen non-business customer flying with Global is on an ordinary jet?

ANSWER:

a. / .357
b. / .583

5. Super Cola sales breakdown as 80% regular soda and 20% diet soda. While 60% of the regular soda is purchased by men, only 30% of the diet soda is purchased by men. If a woman purchases Super Cola, what is the probability that it is a diet soda?

ANSWER:

.30435

6. A food distributor carries 64 varieties of salad dressing. Appleton Markets stocks 48 of these flavors. Beacon Stores carries 32 of them. The probability that a flavor will be carried by Appleton or Beacon is 15/16. Use a Venn diagram to find the probability a flavor is carried by both Appleton and Beacon.

ANSWER:

The Venn diagram is

and P(A B) = P(A) + P(B) - P(A B) = 6/8 + 4/8 - 15/16 = 5/16 = .3125

7. Through a telephone survey, a low-interest bank credit card is offered to 400 households. The responses are as tabled.

Income £ $60,000 / Income > $60,000
Accept offer / 40 / 30
Reject offer / 210 / 120
a. / Develop a joint probability table and show the marginal probabilities.
b. / What is the probability of a household whose income exceeds $60,000 and who rejects the offer?
c. / If income is £ $60,000, what is the probability the offer will be accepted?
d. / If the offer is accepted, what is the probability that income exceeds $60,000?

ANSWER:

a. / Income £ $60,000 / Income > $60,000 / Total
Accept offer / .100 / .075 / .175
Reject offer / .525 / .300 / .825
Total / .625 / .375 / 1.000
b. / .3
c. / .16
d. / .4286

8. A medical research project examined the relationship between a subject's weight and recovery time from a surgical procedure, as shown in the table below.

Underweight / Normal weight / Overweight
Less than 3 days / 6 / 15 / 3
3 to 7 days / 30 / 95 / 20
Over 7 days / 14 / 40 / 27
a. / Use relative frequency to develop a joint probability table to show the marginal probabilities.
b. / What is the probability a patient will recover in fewer than 3 days?
c. / Given that recovery takes over 7 days, what is the probability the patient is overweight?

ANSWER:

a. / Underweight / Normal weight / Overweight / Total
Less than 3 days / .024 / .06 / .012 / .096
3 to 7 days / .120 / .38 / .080 / .580
Over 7 days / .056 / .16 / .108 / .324
Total / .200 / .60 / .200 / 1.00
b. / .096
c. / 27/81 = .33

9. It is estimated that 3% of the athletes competing in a large tournament are users of an illegal drug to enhance performance. The test for this drug is 90% accurate. What is the probability that an athlete who tests positive is actually a user?

ANSWER:

.2177

10. A corporation has 15,000 employees. Sixty-two percent of the employees are male. Twenty-three percent of the employees earn more than $30,000 a year. Eighteen percent of the employees are male and earn more than $30,000 a year.

a. / If an employee is taken at random, what is the probability that the employee is male?
b. / If an employee is taken at random, what is the probability that the employee earns more than $30,000 a year?
c. / If an employee is taken at random, what is the probability that the employee is male and earns more than $30,000 a year?
d. / If an employee is taken at random, what is the probability that the employee is male or earns more than $30,000 a year or both?
e. / The employee taken at random turns out to be male. Compute the probability that he earns more than $30,000 a year.
f. / Are being male and earning more than $30,000 a year independent?

ANSWER:

a. / 0.62
b. / 0.23
c. / 0.18
d. / 0.67
e. / 0.2903
f. / No

Chapter 3—Probability Distributions

1. 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.
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?

ANSWER:

a. / f(x) = 1.25 for 31.8 £ x £ 32.6
= 0 otherwise
b. / 0, .375, 0
c. / no

2. During lunch time, customers arrive at Bob's Drugs according to a Poisson distribution with l = 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?

ANSWER:

a. / .0183, .0733, .1465, .7619
b. / .0107
c. / .8647

3. 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?

ANSWER:

a. / .632
b. / .135
c. / .468

4. 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.

ANSWER:

a. / Relative frequency / Number of tests
.50 / 1
.35 / 2
.10 / 3
.05 / 4
Each probability is between 0 and 1 and the sum is 1.00.
b. / m = 1.7
c. / s 2 = .71
d. / s = .8426

5. 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 / RevenueCost
0 / .35 / 00
1 / .30 / 400
2 / .20 / 800
3 / .10 / 8015
4 / .05 / 8030
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?

ANSWER:

a. / 1.2
b. / 40
c. / 3
d. / 37

6. 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?

ANSWER:

.1840

7. 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?

ANSWER:

a. / 182
b. / P(x = 8 | l = 5.2) = .0731

8. 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

ANSWER:

a. / .0498
b. / .2240
c. / .0500
d. / .3585

9. 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?
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?

ANSWER:

a. / f(x) = 1/3 / for 38.5 £ x £ 41.5
= 0 / otherwise
b. / .5000
c. / .1667
d. / .3333

10. 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?

ANSWER:

a. .3277

b. .2048

c. 230 minutes