# Probability and Business Decisions

a. See the attached file for the chart of data pertaining to this question.

The probabilities shown above represent the states of nature and the decision maker's (e.g., manager) degree of uncertainties and personal judgment on the occurrence of each state. What is the expected payoff for actions A1 and A2 above? What would be your recommendation? Interpret the results based on practical considerations.

b. Bayes and empirical Bayes (EB) methods structure combining information from similar components of information and produce efficient inferences for both individual components and shared model characteristics. For example, city-specific information on the profits involved in selling a particular brand of coffee in Mumbai might be unavailable. How could CoffeeTime "borrow information" from adjacent cities or other countries to employ Bayesian logic?

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#### Solution Preview

The probabilities shown above represent the states of nature and the decision maker's (e.g., manager) degree of uncertainties and personal judgment on the occurrence of each state. What is the expected payoff for actions A1 and A2 above?

Expected payoff = SPi*Xi

Where Pi=Probability of state of nature i

Xi=payoff under state of nature I

Expected payoff A1 = 0.2*3000+0.5*2000+0.3*(-6000) = -200

Expected payoff A2 = 0.2*0+0.5*0+0.3*0 = 0

What would be your recommendation? Interpret the results based on practical considerations.

As the expected payoff is higher when the decision not to sell juices is taken, I recommend a decision for not selling juices.

The practical consideration is that if we decide to ...

#### Solution Summary

This solution illustrates a simple and systematic way to show how decision-making can be done when choices are made under uncertainty. The concepts of probability and expected payoff are used to select the best alternative for the CoffeeTime. The post also discusses the practical considerations along with financial considerations while taking a decision. In the second part, it discusses how the city specific information on the profits can or cannot be used for taking decision in another city

Business Math : Probability, Binomial Distributions, Decision-Making, Payoff Tables and Risk Management

1. Super Cola sales break down as 80% regular soda and 20% diet soda. While 60% of the regular soda is purchased by men, only 30% of the diet soda is purchased by men. If a woman purchases Super Cola, what is the probability that it is a diet soda?

2. A process follows the binomial distribution with n = 8 and p = .3. Find

a. P(x = 3)

b. P(x > 6)

c. P(x  2)

3. Scores on an endurance test for cardiac patients are normally distributed with mean = 182 and standard deviation = 24.

a. What is the probability a patient will score above 190?

b. What percentage of patients score below 170?

c. What score does a patient at the 75th percentile receive?

4. A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained. Following is a record of the number of students in a class of 20 who took the test each number of times.

Students Number of tests

10 1

7 2

2 3

1 4

a. use the relative frequency approach to construct a probability distribution

b. show that it satisfies the required condition for being a probability distribution.

c. Find the expected value of the number of tests taken.

5. (40 points)

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

s1 s2 s3

New Restaurant -$80K $20K $160K

Expand -$40K $20K $100K

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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