# Decision Theory: Probability Questions

1. The weight of a one cubic yard bag of landscape mulch is normally distributed with a mean of 40 pounds and a standard deviation of 2 pounds.

a. What is the probability that a bag will weigh less than 40 pounds?

b. What is the probability that a bag will weigh between 38 and 40 pounds?

2. According to the Spring 2014 issue of the periodical Zero Population Growth, the U.S. leads the fully industrialized world in teen pregnancy rates, with 20% of U.S. females getting pregnant at least once before the age of 18. Suppose that 15 U.S. females are selected at random. Find the probability that the number who has been pregnant at least once before age 18 will be exactly three.

3. It is estimated that 10% of the athletes competing in a large tournament are users of an illegal drug to enhance performance. The test for this drug is 95% accurate. Determine the probability that an athlete who tests positive is actually a user.

4. There are two more assignments in a class before its end, and if you get an A on at least one of them, you will get an A for the semester. Your subjective assessment of your performance is

Event Probability

A on paper and A on exam .25

A on paper only .15

A on exam only .20

A on neither .40

a. What is the probability of getting an A on the exam?

b. What is the probability of getting an A in the course?

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#### Solution Summary

The solution comprises of detailed step-by-step calculations of the given problems on probability.

Decision theory

See attached notes to solve for problems in the attachment. Show work.

3-17

Although Ken Brown is the principal owner of Brown Oil, his brother Bob is credited with making the company a financial success. Bob is vice president of finance. Bob attributes his success to his pessimistic attitude about business and the oil industry. Given the information from problem 3-16, it is likely that Bob will arrive at a different decision. What decision criterion should Bob use, and what alternative will he select?

3-19

Mickey Lawson is considering investing some money that he inherited. The following payoff table gives the profits that would be realized during the next year for each of the three investment alternatives Mickey is considering:

STATE OF NATURE

DECISION GOOD POOR

ALTERNATIVE ECONOMY ECONOMY

Stock Market 80,000 - 20,000

Bonds 30,000 20,000

CDs 23,000 23,000

Probability 0.5 0.5

(a) What decision would maximize expected profits?

(b) What is the maximum amount that should be paid for perfect forecast of the economy?

3-23

Today's Electronics specializes in manufacturing modern electronic components. It also builds the equipment that produces the components. Phyllis Weinberger, who is responsible for advertising the president of Today's Electronics on electronic manufacturing equipment, has developed the following table concerning a proposed facility:

PROFIT ($)

STRONG FAIR POOR

MARKET MARKET MARKET

Large facility 550,000 110,000 - 310,000

Medium-sized facility 300,000 129,000 - 100,000

Small facility 200,000 100,000 - 32,000

No facility 0 0 0

(a) Develop an opportunity loss table.

(b) What is the minimax regret decision?

3-28

A group of medical professionals is considering the construction of a private clinic. If the medical demand is high (i.e., there is a favorable market for the clinic), the physicians could realize a net profit of $100,000. If the market is not favorable, they could lose $40,000. Of course, they don't have to proceed at all, in which case there is no cost. In the absence of any market data, the best the physicians can guess is that there is a 50 - 50 chance the clinic will be successful. Construct a decision tree to help analyze this problem. What should the medical professionals do?

3-29

The physicians in Problem 3-28 have been approached by a market research firm that offers to perform a study of the market at a fee of $5,000. The market researchers claim their experiences enable them to use Bayes' theorem to make the following statements of probability:

probability of a favorable market given

a favorable study = 0.82

probability of an unfavorable market given

a favorable study = 0.18

probability of a favorable market given

an unfavorable study = 0.11

probability of an unfavorable market given

an unfavorable study = 0.89

probability of a favorable research

study = 0.55

probability of an unfavorable research study = 0.45

(a) Develop a new decision tree for the medical professionals to reflect the options now open with the market study.

(b) Use EMV approach to recommend a strategy.

(c) What is the expected value of sample information? How much might the physicians be willing to pay for a market study?

3-32

Bill Holliday is not sure what he should do. He can either build a quadplex (i.e., a building with four apartments), build a duplex, gather information, or simply do nothing. If he gathers additional information, the results could be either favorable or unfavorable, but it would cost him $3,000 to gather the information. Bill believes that there is a 50-50 chance that the information will be favorable. If the rental market is favorable, Bill will earn $15,000 with the quadplex or $5,000 with the duplex. Bill doesn't have the financial resources to do both. With an unfavorable rental market, however, Bill could lose $20,000 with the quadplex or $10,000 with the duplex. Without gathering additional information, Bill estimates that the probability of a favorable rental market is 0.7. A favorable report from the study would increase the probability of a favorable rental market to 0.9. Furthermore, an unfavorable report from the additional information would decrease the probability of a favorable rental market to 0.4. Of course, Bill would forget all of these numbers and do nothing. What is your advice to Bill?

3-41

In this chapter a decision tree was developed for John Thompson. After completing the analysis, John was not completely sure that he is indifferent to risk. After going through a number of standard gambles, John was able to assess his utility for money. Here are some of the utility assessments: U( - $190,000) = 0, U( - $180,000) = 0.15, U(- $30,000) = 0.2, U($0) = 0.3, U($90,000) = 0.5, U($100,000) = 0.6, U($190,000) = 0.95, and U($200,000) = 1.0. If John maximizes his expected utility, does his decision change?