1. A service club has printed and sold 3000 raffle tickets for a painting worth $750. What is the mathematical expectation of a person who holds one of these raffle tickets? 2. As part of a promotional scheme, a soap manufacturer offers a first prize of $3000 and a second prize of $1000 to persons chosen at random from among 15000 persons willing to try a new product and send in their name and address on the label. What is the mathematical expectation of a person taking part in this promotion? 3. An importer pays $12000 for a shipment of bananas, and the probabilities that he will be able to sell them for $16,000, $13,000, $12,000, or only $10,000 are respectively, 0.25, 0.46, 0.19, and 0.10. What is his expected gross profit? 4. To defend a client in a liability suit resulting from a car accident, a lawyer must decide whether to charge a straight fee of $7500 or a contingent fee, which she will only get if she wins. How does she feel about her chances if: a. She prefers a straight fee of $7500 to a contingent fee of $25000? b. She prefers a contingent fee of $60,000 to the straight fee of $7500? 5. A truck driver has to deliver a load of building materials to one of two construction sites, a barn that is 18 miles from the lumberyard, or a shopping center that is 22 miles from the lumberyard. He has misplaced the order indicating where the load should go. Also, he must return to the lumberyard after his delivery. The barn and shopping center are 8 miles apart. To complicate matters, the telephone at the lumberyard is not working. If the truck driver believes the probability is 1/6 that the load should go to the barn and 5/6 that is should go to the shopping center, where should he go first so as to minimize the distance that he will have to drive? 6. With reference to number 5, where should the driver go first so as to minimize the expected driving distance if the probabilities are and instead of 1/6 and 5/6? 7. The ages of the seven entries in an essay contest are 17, 17, 17, 18, 20, 21 and 23, and their chances of winning are all equal. If we want to predict the age of the winner and there is a reward for being right, but none for being wrong, what prediction maximizes the expected reward? 8. The ages of seven entries in an essay contest are 17, 18, 17, 20, 21, 19, and 21, and their chances of winning are all equal. If we want to predict the age of the winner and we make $400 for our prediction minus a penalty fee of 10 times the magnitude of the error squared; what prediction maximizes the amount of money we are expected to make? How much money are we expected to make? 9. In each case determine whether the given values can serve as the values of the probability distribution of some random variable that can take on values 1, 2, and 3, and explain your answers:

a. f(1) = 0.52, f(2)= 0.26, and f(3)= 0.32 b. f(1) = 0.18, f(2)= 0.02, and f(3)= 1.00 c. f(1) = 10/33, f(2)= 1/3, and f(3)=12/33