### MATH 318, Chapter 4 Quiz

Name:
You may use a calculator on this quiz. You may not use a cell phone or computer. Please read each question carefully, show your work and give justifications for your answers. The questions are roughly organized from shortest to longest; if you find that you are spending a lot of time on one problem, leave it blank and move on to the next. There are questions on both sides of this quiz paper.
 P(A | B) = P(A ∩ B) / P(B) E = EVSI / EVPI * 100
1. (20 points) In your own words, what is a state of nature?

2. (20 points) Fictional Textbook Corp. is considering releasing their new book in hardcover format. If they do this, they predict a 20% chance of \$100,000 profit, a 50% chance of \$200,000 profit and a 30% chance of \$500,000 profit. What is the expected value of publishing the new book as a hardcover?

3. (20 points) If the probabilities of the states of nature in the payoff table below are P(S1) = 10%, P(S2) = 60% and P(S3) = 30%, what is the expected value of perfect information (EVPI)?
 S1 S2 S3 d1 -200 300 500 d2 200 400 300 d3 300 200 -100

4. (20 points) Fictional Textbook Corp. is deciding what format to publish their new textbook in. Shown below are the risk profiles for three formats: hardcover, paperback and electronic. Which choice would be recommended by an optomistic approach?

5. (20 points) On a summer day in Plymouth, there is an 80% prior probability of sun and a 20% prior probability of rain. Each day the weather report predicts either "favorable" (sunny) or "unfavorable" (rainy) weather. Assuming the joint probability table shown below is correct, what is the posterior (branch) probability of rain on a day when the weather report is favorable?
 rain sun favorable .04 .72 unfavorable .16 .08 .20 .80