Winning a tennis game from deuce

Question

A tennis game is at deuce. From deuce, a player must go two points ahead to win. You win each point independently with probability p (opponent wins with q = 1 - p). If you win one point you have "advantage"; win the next and you take the game; lose it and you are back to deuce. What is your probability of winning the game from deuce, in terms of p?

Accepted Answer

There are three live states: Deuce (D), Advantage-me (A), Advantage-opponent (B). Let w, a, b be my win probabilities from each. From advantage-me: win the point (prob p) \Rightarrow game won; lose (prob q) \Rightarrow back to deuce: a = p + q\,w. From advantage-opponent: win (prob p) \Rightarrow back to deuce; lose (prob q) \Rightarrow game lost: b = p\,w. From deuce: win (prob p) \Rightarrow A; lose (prob q) \Rightarrow B: w = p\,a + q\,b = p(p + q\,w) + q(p\,w) = p^2 + 2pq\,w. Solve for w: w(1 - 2pq) = p^2 \implies w = \frac{p^2}{1 - 2pq}. Since (p+q)^2 = 1 gives 1 - 2pq = p^2 + q^2, this …

Winning a tennis game from deuce

Your answer

More Probability questions