9.2 Exercise 18

James is trying to decide which of two sales opportunities to take.

• In the first, he makes three independent calls. Payoffs are $310, $380, and $350, with respective probabilities of 0.35, 0.41, and 0.57.
• In the second, he makes eight independent calls, with probability of success on each call $p = 0 . 57$. He realizes $100 profit on each successful sale.

Let X be the net profit on the first alternative and Y be the net gain on the second. Assume the pair $\left\{X , Y\right\}$ is independent.

• Which alternative offers the maximum possible gain?
• What is the probability the second exceeds the first— i.e., what is $P \left(Y > X\right)$?
• Compare probabilities in the two schemes that total sales are at least $600, $700, $750.