Question 1

A large disaster cleaning company estimates that 30 percent of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that it has contracted, calculate the probability that exactly 4 of the jobs were not completed within the bid time.

Accepted Answer

P(x = 4) = .1361

Question 2

According to data from the state blood program, 40 percent of all individuals have group A blood. If six individuals give blood, find the probability that none of the individuals has group A blood.

Accepted Answer

P(x = 0) = .0467

Question 3

An important part of the customer service responsibilities of a cable company is the speed with which service troubles can be repaired. Historically, the data show that the likelihood is .75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day, find the mean number of troubles repaired on the same day.

Accepted Answer

μ_x = np = (5)(.75) = 3.75

Question 4

An insurance company will insure a $75,000 particular automobile make and model for its full value against theft at a premium of $1500 per year. Suppose that the probability that this particular make and model will be stolen is .0075. Find the premium that the insurance company should charge if it wants its expected net profit to be $2000.

Accepted Answer

2000 = (x − 75,000)(.0075) + x(.9925) = .0075x − 562.5 + .9925x = 2562.5

Question 5

An important part of the customer service responsibilities of a cable company is the speed with which service troubles can be repaired. Historically, the data show that the likelihood is .75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that at least three troubles will be repaired on the same day?

Accepted Answer

P(x ≥ 3) = 1 − (P ≤ 2) = .8965

Question 6

According to data from the state blood program, 40 percent of all individuals have group A blood. Suppose that of six randomly selected individuals, three have group A blood. Would you believe the data from the state blood program?

Accepted Answer

Yes, the probability is greater than 0.05, and therefore we do not reject the null hypothesis.

Question 7

An important part of the customer service responsibilities of a cable company is the speed with which service troubles can be repaired. Historically, the data show that the likelihood is .75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that all five will be repaired on the same day?

Accepted Answer

P(x = 5) = .2373

Question 8

A lawyer believes that the probability is .3 that she can win a discrimination suit. If she wins the case, she will make $400,000; but if she loses, she gets nothing. Assume that she has to spend $75,000 preparing the case. What is her expected gain?

Accepted Answer

μ_x = .7(−75,000) + .3(325,000) = −52,500 + 97,500 = 45,000

Question 9

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day. When there are 8 local letters mailed, what is the probability that the number delivered will be within 2 standard deviations of the mean?

Accepted Answer

P[σ = 7.2 ± 2(.85)] = P(σ = 7.2 ± 1.7) = P(5.5 ≤ x ≤8) = P(6 ≤ x ≤8)
= .4305 + .3826 + .1488 = .9619

Question 10

An important part of the customer service responsibilities of a cable company is the speed with which service troubles can be repaired. Historically, the data show that the likelihood is .75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that fewer than two troubles will be repaired on the same day?

Accepted Answer

P(x < 2) = P(x = 0) + P(x = 1) = .0156

Question 11

The internal auditor for your company believes that 10 percent of your invoices contain errors. To check this theory, 20 invoices are randomly selected, and 5 are found to have errors. What is the probability that of the 20 invoices selected, 5 or more would contain errors if the theory is valid?

Accepted Answer

P(x ≥ 5) = .0319 + .0089 + .0020 + .0004 + .0001 = .0433

Question 12

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day. If you mail eight local letters, what is the average number you expect to be delivered the next day?

Accepted Answer

μ_x = np = (8)(.9) = 7.2

Question 13

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day. Calculate the standard deviation of the number delivered when 8 local letters are mailed.

Accepted Answer

σ ==== .85

Question 14

An insurance company will insure a $75,000 particular make and model of car for its full value against theft at a premium of $1,500 per year. Suppose that the probability that this particular automobile make and model will be stolen is .0075. Calculate the expected net profit for the insurance company.

Accepted Answer

μ_x = (−73,500)(.0075) + (1500)(.9925) = −551.25 + 1488.75 = 937.50

Question 15

According to data from the state blood program, 40 percent of all individuals have group A blood. If six individuals give blood, find the probability that at least 3 of the individuals have group A blood.

Accepted Answer

P(x ≥ 3) = p(x = 3) + p(x = 4) + p(x = 5) + p(x = 6) = .4557

Question 16

A car wash loses $30 on rainy days and makes $120 on days when it does not rain. If the probability of rain is .15, calculate expected profit for the car wash.

Accepted Answer

μ_x = (−30)(.15) + (120)(.85) = −4.50 + 102 = 97.50

Question 17

According to data from the state blood program, 40 percent of all individuals have group A blood. If six individuals give blood, find the probability that exactly three of the individuals have group A blood.

Accepted Answer

P(x = 3) = .2765

Question 18

According to data from the state blood program, 40 percent of all individuals have group A blood. If six individuals give blood, find the mean number of individuals having group A blood.

Accepted Answer

μ_x = np = (6)(.4) = 2.4

Question 19

A large disaster cleaning company estimates that 30 percent of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that it has contracted, calculate the mean number of jobs completed within the bid time.

Accepted Answer

μ_x = np = 8(.3) = 2.4

Question 20

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day. If you mail eight local letters, what is the probability that all of them will be delivered the next day?

Accepted Answer

P(x = 8) = .4305

Quiz 6: Discrete Random Variables