"Game, Set, Match": Double Delight Watching a Grand Slam Tennis Match

Abstract

Probabilistic properties of tennis scoring systems are examined and compared with best-of-K systems. A model, where each player has his/her own probability of winning his/her service point and which remains invariant for the duration of the match, and where outcomes of points played are independent of each other, is assumed. Probabilities of winning a game tie-breaker, a game, a set tie-breaker, a set, and the match are obtained. Since tennis scoring systems are unique, probability calculations require decomposing big and complicated problems into smaller and simpler constituent problems, solving these sub-problems, then combining to obtain the solution to the big problem. The problems that arise from tennis scoring systems offer excellent pedagogical venues for teaching probability, in particular, the use of the Theorem of Total Probability and the Iterated Rules for Mean, Variance, and Covariance. There are also many interesting questions in tennis, foremost of which is whether a tennis match under this assumption will actually end with probability one; or whether when two players of `equal abilities' play a match, the first server possesses an advantage. These questions are addressed in this work. Tennis scoring systems are technically statistical decision systems to determine the better player. Since such a decision system is based on a finite number of points played, erroneous decisions could arise, such as the inferior player winning the match. We compare different systems in terms of the probability of the better player winning, as well as the duration of the match in terms of the number of points played.

Figures (19)

• Figure 1: Carlos Alcaraz of Spain serving against Arthur Rinderknech of France in a Round-of-16 match at the US Open in Arthur Ashe Stadium, Flushing, New York City, August 31, 2025.
• Figure 2: Characteristics of a Game. Top left plot is the probability of server winning the game with respect to server's probability of winning point; top right plot is the PMF of number of points played when $p=.5$. Bottom plots are the means and standard deviations of the number of points for $p \in (0,1)$.
• Figure 3: Tree diagram for the set tie-breaker tie-breaker. (i) If the serving sequence follows $ABBAABB \ldots$. (ii) If the serving sequence follows alternate serve ($ABABAB \ldots$).
• Figure 4: Contour plot of the probability of first server winning the set tie-breaker when $K = 7$. $pA$ is the probability of first server winning point on his/her serve; while $pB$ is the probability of the first receiver winning point on his/her serve.
• Figure 5: Probability mass function (pmf) of $N_{ST}$ for $K=7$ and when $p_A=p_B=.5$ (left) and $p_A=p_B=.9$ (right).

Theorems & Definitions (32)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof