# Probability Problems

1) A shipping company knows that the cost of delivering a small package within 24 hours is $14.80. The company charges $15.50 for shipment, but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 2% of its packages within the 24 hour period, what is the expected gain per package?

2) A study of the behavior of a large number of drug offenders after treatment for drug abuse suggests that the likelihood of conviction within a 2 year period after treatment may depend on the offender's education. The proportion of the total number of cases that fall into 4 Education/Conviction categories are shown in the table below:

Status within 2 years After Treatment

Education Convicted Not Convicted Totals

10yrs or more .10 .30 .40

9yrs or less .27 .33 .60

Totals .37 .63 1.00

Suppose a single offender is selected from the treatment program. Here are the events of interest:

A. The offender has 10 or more years of education.

B. The offender is convicted within 2 years after completion of treatment.

Find the appropriate probabilities for these events:

1) A

2) B

3) A intersection B

4) A Union B

5) A Complement

6) Complement of A union B

7) Complement of A Intersection B

8) A given that B has occurred

9) B given that A has occurred

3) Accident records collected by an automobile insurance co. give the following information:

The probability that an insured driver has an automobile accident is .15; If an accident has occurred, the damage to the vehicle amounts to 20% of its market value with probability .80, 60% of its market value with probability .12, and a total loss with probability .08.

What Premium should the Insurance Company charge on a $22,000 car so that the expected gain by the company is 0?

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#### Solution Preview

1) A shipping company knows that the cost of delivering a small package within 24 hours is $14.80. The company charges $15.50 for shipment, but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 2% of its packages within the 24 hour period, what is the expected gain per package?

Profit made when package delivered= $0.70 =15.5-14.8

Loss made when package not delivered= -$14.80 =Cost

Probability Profit/loss Prob X Profit

package delivered 0.98 $0.70 $0.69

package not delivered 0.02 -$14.80 -$0.30

Expected gain = Total= $0.39

Expected gain per package= $0.39

2) A study of the behavior of a large number of drug offenders after treatment for drug abuse suggests that the likelihood of conviction within a 2 year period after treatment may depend on the offender's education. The proportion of ...

#### Solution Summary

There are answers to 3 questions on probability