1. 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 .10 A on exam only .30 A on neither .35 Let's develop a joint-probability table to analyze this problem (the letters a – h represent probability values. For example, a is the probability of getting an A on both the paper and the exam; b is the probability of getting an A on the exam, but not on the paper; while g is the sum of a and d, and represents the probability of getting an A on the paper). Get A on paper Do not get A on paper row sum Get A on exam a b c Do not get A on exam d e f column sum g h a. What does c + f equal? b. What does a + b + d + e equal? c. What is the probability of getting an A on the exam? d. What is the probability of getting an A in the course? e. Are the grades on the assignments independent (explain your answer mathematically)? 2. 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? 3. The Ambell Company uses batteries from two different manufacturers. Historically, 60% of the batteries are from manufacturer 1, and 90% of these batteries last for over 40 hours. Only 75% of the batteries from manufacturer 2 last for over 40 hours. A battery in a critical tool fails at 32 hours. What is the probability it was from manufacturer 2? 4. The high school GPA of applicants for admission to a college program are recorded and relative frequencies are calculated for the categories. GPA f(x) x < 2.0 .08 2.0 <= x < 2.5 .12 2.5 <= x < 3.0 .35 3.0 <= x < 3.5 .30 3.5 <= x ? a. Complete the table to make this a valid probability distribution. b. What is the probability an applicant's GPA will be below 3.0? c. What is the probability an applicant's GPA will be 2.5 or above? 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 tape rental. Demand Relative Frequency Revenue Cost 0 .35 0 0 1 .30 40 0 2 .20 80 0 3 .10 80 15 4 .05 80 30 a. What is the expected demand? b. What is the expected revenue? c. What is the expected cost? d. What is the expected profit? 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? (note: answer using either the relevant probability table in the back of the book, or use the relevant probability function on your calculator / Excel). 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? 8. The weight of a can of stewed tomatoes is listed as 16 oz., but in actuality, the weight between cans may vary, and in fact is normally distributed with mean = 16.2 ounces and standard deviation = 0.1 o

8. The weight of a can of stewed tomatoes is listed as 16 oz., but in actuality, the weight between cans may vary, and in fact is normally distributed with mean = 16.2 ounces and standard deviation = 0.1 oz. What is the probability that a consumer buys a can whose contents are actually less than the stated weight of 16 oz.?

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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

.10

A on exam only

.30

A on neither

.35

Let's develop a joint-probability table to analyze this problem (the letters a – h represent probability values. For example, a is the probability of getting an A on both the paper and the exam; b is the probability of getting an A on the exam, but not on the paper; while g is the sum of a and d, and represents the probability of getting an A on the paper).

Get A on paper

Do not get A on paper

row sum

Get A on exam

a

b

c

Do not get A on exam

d

e

f

column sum

g

h

What does c + f equal?

What does a + b + d + e equal?

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

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

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1. The options from which a decision maker chooses a course of action are a. called the decision alternatives. b. under the control of the decision maker. c. not the same as the states of nature. d. each of the above is true.

2. States of nature a. can describe uncontrollable natural events such as floods or freezing temperatures. b. can be selected by the decision maker. c. cannot be enumerated by the decision maker. d. each of the above is true.

3. Which of the methods for decision making without probabilities best protects the decision maker from undesirable results? a. the optimistic approach b. the conservative approach c. minimum regret d. minimax regret

4. If P(high) = .3, P(low) = .7, P(favorable | high) = .9, and P(unfavorable | low) = .6, then P(favorable) = a. .10 b. .27 c. .30 d. .55

5. If the payoff from outcome A is twice the payoff from outcome B, then the ratio of these utilities will be a. unknown without further information. b. less than 2 to 1. c. more than 2 to 1. d. 2 to 1

6. The probability for which a decision maker cannot choose between a certain amount and a lottery based on that probability is a. the utility probability. b. the lottery probability. c. the uncertain probability. d. the indifference probability.

7. When the decision maker prefers a guaranteed payoff value that is smaller than the expected value of the lottery, the decision maker is a. an optimizer. b. a risk taker. c. an optimist. d. a risk avoider.

8. A decision maker whose utility function graphs as a straight line is a. a risk avoider. b. risk neutral. c. a risk taker. d. conservative

9. When consequences are measured on a scale that reflects a decision maker's attitude toward profit, loss, and risk, payoffs are replaced by a. opportunity loss. b. multicriteria measures. c. sample information. d. utility

10. The purchase of insurance and lottery tickets shows that people make decisions based on a. maximum likelihood b. sample information. c. utility. d. expected value.

11. The expected utility approach a. requires a decision tree. b. leads to the same decision as the expected value approach. c. is most useful when excessively large or small payoffs are possible. d. does not require probabilities.

12. Values of utility a. must be between 0 and 1. b. must be between 0 and 10. c. must be nonnegative. d. must increase as the payoff improves.

II. PROBLEMS (each question is worth 32 pts – 8 pts. each part)

1. A payoff table is given as

s1 s2 s3 d1 10 8 6 d2 14 15 2 d3 7 8 9

a. What decision should be made by the optimistic decision maker? b. What decision should be made by the conservative decision maker? c. What decision should be made under minimax regret? d. If the probabilities of s1, s2, and s3 are .2, .4, and .4, respectively, then what decision should be made under expected value?

2. Chez Paul is contemplating either opening another restaurant or expanding its existing location. The payoff table for these two decisions is:

Paul has calculated the indifference probability for the lottery having a payoff of $160K with probability p and -$80K with probability (1-p) as follows:

Amount Indifference Probability (p) -$40K .4 $20K .7 $100K .9

a. Is Paul a risk avoider, a risk taker, or risk neutral? EXPLAIN.

b. Suppose Paul has defined the utility of -$80K to be 0 and the utility of $160K to be 80. What would be the utility values for -$40K, $20K, and $100K based on the indifference probabilities?

c. Suppose P(s1) = .4, P(s2) = .3, and P(s3) = .3. Which decision should Paul make using the expected utility approach?

d. Compare the result in part c with the decision using the expected value approach.

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