In a match of women’s tennis, the winner is the player who first wins two sets.

Suppose two players, Martina and Steffi, play against each other and that the probability that Martina wins any particular set is p for some 0 < p < 1.

(i) Find, as a function of p, the probability that Martina wins the match.

(ii) Let Y be the number of sets to be played. Find E(Y).

(iii) Show that the answer in (ii) is maximized when p = 1∕2.