# Chasing The Ace

## Why it pays to have a strong serve in tennis.

29 June 2022

Wimbledon 2010 saw one of the most freakish sporting contests of all time.

In a match that spanned three days, John Isner (USA) finally broke the serve of his opponent Nicolas Mahut (France) to win the final set by 70 games to 68. It smashed the previous Wimbledon fifth set record, which had been 23-21.

But how could this extraordinary result have come about? What were the chances?

It turns out that the chances were a lot higher than you might expect. It all happened because both players' serves were too strong.

Let's do the maths.

Suppose you have a super-power which means every time you serve it's an ace. You always win every point you play. Your chance of winning your service game is of course 100%.*

What if instead you have a feeble serve, always serving double faults? This time you win 0% of your service games.

But in real matches, no server is this good or bad. The server will lose at least some of the points in a game. Let's suppose your serve is nothing special (like mine!), and that on average you only win 50% of the points on your serve, almost as if you were flipping a coin. The chance of winning a service game is now 50%. If this isn't obvious, think about your opponent, who also has a 50% chance of winning a point. Since server and opponent have the same odds of success in a point, it must be the case that each has the same 50% chance of winning a game.

So we now know that if the probability of winning a point is 0%, 50% or 100%, the corresponding chance of winning a game is also 0%, 50% and 100%, Plotted on a graph, they form a straight line. So, if your chance of winning a point is 80%, does this mean your chance of winning the game will also be 80%? No!!

It turns out that if your chance of winning any service point is 80%, your chance of winning that service game is about 98% - you are almost certain to win the game. Which at first glance seems...surprising. The graph of your chance of winning a game plotted against the chance of you winning a point on your serve is not a straight line. It is an S-shaped curve.

The calculations for this are quite messy, but perhaps you can see why the result is true by thinking about what happens when you play lots of points. After one point your chance of being ahead of your opponent is 80%. But imagine an extreme in which a game requires you to be the first to win 100 points. On average, the non-server will only be on 25 points when the server gets to 100, and it would take an almost unimaginable sequence of luck for the non-server to over-turn that lead.

The problem for Isner and Mahut is that both were winning too high a proportion of their service points, and even an 80% success rate on each point meant that they were each almost certain to win every one of their service games.

In 2019 Wimbledon finally solved the long match problem by introducing a tie break in the fifth set. That set could now only go to 13-12 (as indeed it did in 2019 when Djokovic beat Federer). In 2022 the authorities curtailed the fifth set even further, introducing a sudden death ten point tie break if the score reaches 6-6.

Meanwhile, your own tennis game might benefit here. A small improvement in your average tennis serve point success from (say) 50% to 55% will convert to a much higer chance of you winning your service game. It's advice that I'll be taking on board this summer.

** A reminder of the scoring: you need to win a minimum of four points to win a game, the first three of which for historic reasons are scored 15, 30 and 40.*

THIS BLOG POST IS ADAPTED FROM THE TENNIS CHAPTER IN ** THE HIDDEN MATHS OF SPORT**, WHICH I WROTE WITH JOHN HAIGH.

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