From Points to Matches: The Probability Math Behind Tennis

Tennis rewards small edges. A player who wins only 55 points out of every 100 on a given serve does not win 55% of games. Because the scoring system stacks points into games, games into sets, and sets into matches, small advantages compound quickly. The math is elegant and surprisingly easy to follow.

From a point to a game

Assume a player wins each point on serve with probability p, independently. To win a game you need four points and a two-point lead. Without the deuce rule this is a binomial race, and the binomial distribution calculator handles the first few points. With deuce, the probability of winning the game is p⁴(1 + 4q + 10q²) + 20p⁵q³ / (1 − 2pq), where q = 1 − p.

The results are striking. A point probability of 0.55 gives a game probability of about 62%, 0.60 gives 74%, 0.65 gives 83% and 0.70 gives 90%.

A modest point edge becomes a large game edge, because the winner must lead twice.

From a game to a set

Now suppose a player won each game with those probabilities and the set needed six games with a two-game margin, with a tiebreak at 6-6. A 62% game probability becomes about an 81% chance of winning the set, and a 74% game probability becomes about 96%. The same compounding repeats.

For a best-of-three match, the chance of taking two sets is s²(3 − 2s), where s is the set probability. Using those numbers, a 55% point edge becomes about 91% for the match, and a 60% edge becomes essentially certain. The geometric distribution calculator and the two-events probability calculator help build intuition for these waiting-and-combining questions.

What the model assumes

Newton and Keller derived formulas of this kind, assuming points are independent and identically distributed (Newton and Keller, 2005). Real tennis is messier. Analysis of almost 90,000 points at Wimbledon found that points are neither independent nor identically distributed, since winning the previous point helps a little and important points are harder for the server. The deviations were small enough that the simple model remains a good approximation (Klaassen and Magnus, 2001).

Our shortcut also treats every game as equally likely to be won. In real matches players alternate serving, and a server's point probability is higher than a returner's. The structure is the same, but the numbers shift.

Why upsets still happen

A 55% point edge still leaves about a 9% chance that the weaker player wins a best-of-three match, which is why upsets are routine even in a sport this lopsided.

Does serving first matter?

One surprising result of the model is that the probability of winning a set, and therefore a match, does not depend on who serves first. Newton and Keller gave an explicit proof of this under the independence assumption. The intuition is that the server alternates, so both players serve equally often within a set, apart from the tiebreak details.

Tiebreaks add one more twist. A set that reaches 6-6 is decided by a short race with its own scoring, which effectively resets the compounding. The calculation above treats it as a single game, an approximation that is good enough for intuition.

It also echoes the lesson from our article on penalty shootouts: small structural edges are hard to detect in few events.