Expected length and probability of winning a tennis game

Abstract

The game of tennis has provided mathematicians with many interesting problems. In [1], the problem of finding the probability that a certain player wins a tennis tournament was studied. Gale [2] determined the best serving strategy in tennis. First, we assume Alice and Bob play a game of tennis using the standard (or Deuce/Ad) scoring system, without a tiebreaker, and that Alice serves the game. We also assume that the probability that Alice wins any point she serves is . Stewart [3] proved that the probability that Alice wins is $${{15{p^4} - 34{p^5} + \;28{p^6} - 8{p^7}} \over {1 - 2p\; + \;2{p^2}}}$$ .