An Analysis of Classical and Tie‐breaker Tennis

Abstract

Expressions for the probability that a player wins a game, set or match of classical or tie-breaker tennis are obtained. Also, expressions for the distribution, the mean and the variance of the number of points in a game, set or match of classical or tie-breaker tennis are obtained. These results explain the long matches often previously observed in classical tennis between two players each with very effective serves on fast grass-court surfaces. The methodology used to derive these expressions could be used to obtain corresponding expressions for other "nested-type'' scoring systems. For example, tennis is 3-nested (game, set, match) whereas squash is only 2-nested (game, match).

Measuring excitement in sport

“…points have been played in the tiebreak game, starting with player A serving on the first point. The Table 5 The excitement of various points within the tiebreak game when 0.2191 (6,7,b) (also (8,9,b), (10,11,b), etc) 0.2922 (7,7,b) (also (9,9,b), (11,11,b), etc) 0.2191 (7,8,a) (also (9,10,a), (11,12,a), etc) 0.2557 total excitement in this tiebreak game conditional on player A winning the game (with probability 0.6630 (in agreement with Pollard (1983))) has a mean of 1.2614 and a variance of 0.6074. The total excitement conditional on player A losing the tiebreak game (with probability 0.3370) has a mean of 1.6305 and a variance of 0.5290.…”

“…The total game-excitement for a (4-point) game with outcome 15-0, 30-0, 40-0, game (won by player A) is, for example, 0.1389, and the total game-excitement for the (12-point) game with outcome 0-15, 0-30, 15-30, 15-40, 30-40, deuce, Ad R, deuce, Ad R, Deuce, Ad S, game (won by A) is 2.3818. Overall, the total excitement on one of player A's service games conditional on him winning the game (with probability 0.9008 (in agreement with Pollard (1983))) has a mean of 0.4414 and a variance of 0.1841. The total excitement on one of player A's service games conditional on him losing the game (with probability 0.0992) has a mean of 1.0268 and a variance of 0.1875.…”

“…The total excitement on one of player B's service games conditional on player A winning the game (with probability 0.2643) has a mean of 1.2836 and a variance of 0.3914. The total excitement on one of player B's service games conditional on player A losing the game (with probability 0.7357 (in agreement with Pollard (1983))) has a mean of 0.8998 and a variance of 0.4100.…”

“…Thus, the total excitement in a tiebreak set of tennis (with pa = 0.7 and pb = 0.6) has a mean of 2.1008 and a variance of 2.1178. Also, as a check in the spreadsheet it was noted that the probability player A wins the tiebreak set is equal to 0.7949 (in agreement with Pollard (1983)). Table 7 gives the mean and variance of the total set excitement (TSE) conditional on the set score.…”

A General Measure for the Relative Efficiency of Any Two Scoring Systems

“…This alternative approach does not depend on there being a ' (P, µ, n) Miles (1984) noted that efficiency under nesting was 'roughly multiplicative ' (p. 107). An aspect of this approximation is that the expected number of points in a set of tennis is only approximately equal to the expected number of points in a game (different for each player) multiplied by the expected number of games in a set (Pollard, 1983). Further, the expected number of service games for one player is typically different to the expected number for the other.…”