1. John has a choice between two options, to play 3 tennis sets with his father (”F”) and Bill (”B”), a
tennis champion. He can either play in the order F-B-F, or in the order B-F-B. Bill is a better player
than his father. If John wins two sets in a row, he will win a prize.
(a) Which order should John choose? Why?
(b) Let Z be the number of times John wins. If, instead, John wanted to maximize E[Z], should
he make a different choice? Why or why not?
2. Suppose X is a binomial random variable with parameters n and p. (i.e., there are n trials). Suppose
Y is a binomial random variable with parameters n and q. Suppose also that 0 p < q 1. True
or false?: P(X k) P(Y k), for any k. If true, provide a proof. If not true, provide a
counterexample. (Hint: if you’re having trouble, simplify by considering small n.)