9.2 Exercise 16

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

• In the first, she makes three independent calls. Payoffs are $570, $525, and $465, with respective probabilities of 0.57, 0.41, and 0.35.
• In the second, she makes eight independent calls, with probability of success on each call $p = 0 . 57$. She realizes $150 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.

1. Which alternative offers the maximum possible gain?
2. Compare probabilities in the two schemes that total sales are at least $600, $900, $1000, $1100.
3. What is the probability the second exceeds the first— i.e., what is $P \left(Y > X\right)$?