# Probability and Decision Analysis

1. Use the normal approximation without the correction factor to find the probabilities in the following exercises.

a. In a binomial experiment with n = 300 and p=.5, find the probability that P is greater than 60%.

b. Repeat Part a with p = 55

c. Repeat Part a with p = .6

2. Determine the probability that in a sample of 100 the sample proportion is less than .75 if p=.80.

3. A commercial for a manufacturer of household appliances claims that 3% of all its products require a service call in the first year. A consumer protection association wants to check the claim by surveying 400 households that recently purchased one of the company's appliances. What is the probability that more than 5% require a service call within the first year? What would you say about the commercial's industry honesty if in a random sample of 400 households 5% report at least one service call?

4. The Red Lobster restaurant chain regularly surveys its customers. On the basis of these surveys, the management of the chain claims that 75% of its customers rate the food as excellent. A consumer rate the food as excellent. A consumer testing service wants to examine the claim by asking 460 customers to rate the food. What is the probability that less than 70% rate the food as excellent?

5. A baker must decide how many specialty cakes to bake each morning. From past experience, she knows that the daily demand for cakes ranges from 0 to 3. Each cake costs $3.00 to produce and sells for $ 8.00 and any unsold cakes are thrown into the garbage at the end of the day.

a. Set up a payoff table to help the baker decide how many cakes to bake.

b. Set up the opportunity loss table.

c. Draw the decision tree.

6. Refer to Problem 5. Assume that the probability of each value of demand is the same for all possible demands.

a. Determine the EMV decision

b. Determine the EOL decision

7. The manager of a larger shopping center in Buffalo is in the process of deciding on the type of snow clearing service to hire for his parking lot. The two services are available. The White Christmas Company will clear all snowfalls for a flat fee of$40,000 for entire winter season. The Weplowen Company charges $18,000 for each snowfall it clears. Set up the payoff table to help the manager decide, assuming that the number of snowfalls per winter season ranges from 0 to 4.

8. Refer to Problem 7. Using subjective assessments, the manager has assigned the following probabilities to the number of snowfalls. Determine the optimal decision.

P(0) = .05 P(1) = .15 P(2) = .30 P(3)=.40 P(4)=.10

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#### Solution Summary

This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of the underlying concepts.

Decision Analysis and Probability Tree

Problems: 9-13, 15, 16, 18, 19, 20

Scenario 15-1

An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.

A B C D

1 Payoff Matrix

2

3 Economy

4 Investment Decline Expand

5 A 0 85

6 B 25 65

7 C 40 30

8 Bank 10 10

Payoffs

____ 9. Refer to Scenario 15-1. What decision should be made according to the maximin decision rule?

a. A

b. B

c. C

d. Bank

____ 10. Refer to Scenario 15-1. What decision should be made according to the minimax regret decision rule?

a. A

b. B

c. C

d. Bank

Scenario 15-2

An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.

A B C D E F G H

1 Payoff Matrix Regret Matrix

2

3 Economy Economy

4 Investment Decline Expand Investment Decline Expand

5 A 0 85 A

6 B 25 65 B

7 C 40 30 C

8 Bank 10 10 Bank

____ 11. Refer to Scenario 15-2. What formula should go in cell F5 of the Regret Matrix to compute the regret value?

a. =B$5-MAX(B$5:B$8)

b. =MAX(B$5:B$8)-MAX(B5)

c. =MAX(B$5:B$8)-MIN(B$5:B$8)

d. =MAX(B$5:B$8)-B5

____ 12. Expected regret is also called

a. EMV.

b. EOL.

c. EPA.

d. EOQ.

Scenario 15-3

An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 70% and an expanding economy at 30%.

A B C D

1 Payoff Matrix

2

3 Economy

4 Investment Decline Expand EMV

5 A -10 90

6 B 20 50

7 C 40 45

8 Bank 15 20

9

10 Probability 0.7 0.3

Payoffs

____ 13. Refer to Scenario 15-3. What decision should be made according to the expected monetary value decision rule?

a. A

b. B

c. C

d. Bank

Scenario 15-5

An investor is considering 4 investments, A, B, C, D. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following decision tree has been developed for the problem. The investor has estimated the probability of a declining economy at 40% and an expanding economy at 60%.

____ 15. Refer to Scenario 15-5. What is the correct decision for this investor based on an expected monetary value criteria?

a. A

b. B

c. C

d. D

____ 16. Refer to Scenario 15-5. What is the expected monetary value for the investor's problem?

a. 32

b. 36

c. 38

d. 42

Scenario 15-6

A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is an 69% chance that demand for their products will be high and a 31% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 63% chance that the customers will like the products and a 37% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is a 92% chance that demand will be high for the products. If the survey is unfavorable there is only a 30% chance that the demand will be high. The following decision tree has been built for this problem. The company has computed that the expected monetary value of the best decision without sample information is 154.35 million. The company has developed the following conditional probability table for their decision problem.

A B C D

1

2 Joint Probabilities

3 High Demand Low Demand Total

4 Favorable Response 0.58 0.05 0.63

5 Unfavorable

Response 0.11 0.26 0.37

6 Total 0.69 0.31 1.00

7

8

9 Conditional Probability

10 For A Given Survey Response

11 High Demand Low Demand

12 Favorable Response 0.92 0.08

13 Unfavorable Response 0.30 0.70

14

15 Conditional Probability

16 For A Given Demand Level

17 High Demand Low Demand

18 Favorable Response 0.84 0.16

19 Unfavorable Response 0.16 0.84

____ 18. Refer to Scenario 15-6. What formula should go in cell C13 of the probability table?

a. =C5/$D4

b. =C5/C$6

c. =C5/$D5

d. =C4/$D4

Scenario 15-7

A decision maker is faced with two alternatives. The decision maker has determined that she is indifferent between the two alternatives when p=0.45.

Alternative 1: Receive $82,000 with certainty

Alternative 2: Receive $143,000 with probability p and lose $15,000 with probability (1-p).

____ 19. Refer to Scenario 15-7. What is the decision maker's certainty equivalent for this problem?

a. -$15,000

b. $82,000

c. $56,100

d. $82,000

____ 20. What is the formula for the weighted average score for alternative j when using a multi-criteria scoring model?

a.

b.

c.

d.