Exercise in probability and statistics, or the probability of winning at tennis

The relationships between the probabilities p, x, s, and M, of winning, respectively, a point, a game, a set, or a match have been derived. The calculations are carried out under the assumption that these probabilities are averages. For example, x represents an average probability of winning a game when serving and receiving, and the same value of x is assumed to hold also for tie‐break games. The formulas derived are for sets played with a tie‐break game at the level of 6–6, as well as for the traditional rule requiring an advantage of two games to win a set. Matches to the best of three and five sets are considered. As is to be expected, a small advantage in the probability p of winning a point leads to advantages which are amplified by large factors : 2.5 for games, 7.1 for sets with tie‐break at 6–6, 10.6 for matches to the best of three sets, and 13.3 for matches to the best of five sets. When sets are decided according to the traditional rule, the last three factors become, respectively, 7.4, 11.1, ...