Suppose you are serving at tennis, and your probability of winning each serve is p. What’s the probability P that you’ll win this game?

Well, let W(n) be the probability of winning the game at serve n. The possible values are n = 4, 5, 6, 8, 10, 12,…. Let T(n) be the probability that the score is tied after serve n, and define s = 2p(1-p).

We calculate W(4) = p⁴, W(5) = 4(1-p)p⁴ = 2p³s, W(6) = 10p⁴(1-p)² = (5/2)p²s², and T(6) = (5/2)s³. After this, the pattern is completely regular. For any k>0, we have T(6+2k) = sT(4+2k) = s^kT(6) = (5/2)s^k+3 and W(6+2k) = p²T(4+2k) = (5/2)p²s^k+2. So W(8)+W(10)+W(12)+… = (5/2)p²s³/(1-s).

Grand total: P = p⁴+2p³s+(5/2)p²s²+(5/2)p²s³/(1-s). With a bit of work this simplifies to P = 1/2 + (10r-5r³+4r⁵–r⁷)/(8+8r²) with r = 2p-1.

For example, if you win 2/3 of your serves, your probability of winning the game is 208/243.

[13 July 2023]