A deuce in a tennis match is when both players are tied at three “points” each. If you ever watch tennis, when the score is a deuce, the player that wins the next point is said to have the advantage, in that if they score the next point, they win the game. If they lose the next point, the score is a deuce again. Thus, in order to win the game, a player must win the next two rallies.
Suppose players A and B are currently tied. Let W be the event that player A wins the game, T be the event that the game ends after two rallies, and D be the event that it becomes a deuce again. If one were to assume that the outcome of each rally is independent of others, and the probability that player A wins any one rally is p,
Show that P(W) = p^2 + 2p(1 ? p) · P(W|D) and use P(W) = P(W|D) to determine P (W ). Briefly explain why P (W ) = P (W |D) must be true.