Game, set and maths

It’s tennis time again. True, tennis time was very short for the Australian men at the French Open, and the Australian women didn't survive too much longer. So, we'll be biding our time until the inevitable and mouth-watering Nadal-Djokovic Final. And, as we’ve done before, we’d like to engage in some tennis maths while we’re waiting.

Every tennis fan knows that the strategy and excitement of a professional match revolves around the service breaks. So, just how valuable is the serve in tennis? Even on the slow clay courts of the French Open, it’s a distinct advantage. 

There were 13975 points in round 1 of this year’s Men’s Singles, and 8745 of these points were won by the server. That suggests that the chances of the server winning a given point are 63%, or about 5/8.

How about the chances of winning a game? There were 2181 games in round 1, of which 1676 were won by the server. That amounts to about 77%, a little over 3/4 of the games.

That was fun, but Nadal and Djokovic are still not on court. So, we’ll while away some more time with a further question, one that only a mathematically inclined tennis fan might ask: are the probabilities 5/8 and 3/4 that we calculated above consistent? 

To make clear what we’re asking, let’s suppose Robin Söderling is serving with Andy Roddick receiving. If Söderling really has a 5/8 chance of winning any given point, does he then have a probability of about 3/4 of winning each game? Let’s see.

Ignoring the quaint scoring names, a game of tennis is the race to win 4 points. However, if the game gets to 3-3, or “deuce”, then a player has to win by two points. We can now employ our trusty (unsmashed) calculator to figure out the odds.

Söderling’s chances of winning 4-0 is just 5/8 multiplied by itself four times, which comes to 625/4026, or about 15%. Similarly, Roddick’s chances of winning 0-4 is 81/4096, about 2%.

We can similarly calculate the other possible results. For example, there are four different ways that Söderling might win 4-1, each with a chance of about 5.7%. In total, that makes Söderling’s chances of winning 4-1 about 23%.

In this manner, and ably assisted by the Binomial Theorem and Pascal’s Triangle, we can easily calculate all the odds. These are indicated in the table below.

However, what about the final column, the 26% of games where Söderling and Roddick reach deuce? Many things, in fact infinitely many things, can now happen. The longest known professional game consisted of 46 points, and theoretically there is no limit to the length of a game. We have to somehow add up the odds for these infinitely many possibilities.

Luckily, infinity doesn’t frighten mathematicians. (Well, only slightly). It is not too difficult to sum the infinite possibilities as what is known as a geometric series. However, there is also an easier approach. (The following idea was also the key to solving the tennis puzzle in our very first Maths Challenge) 

Suppose the game is at deuce, and we’ll let W stand for Söderling’s chances of winning from there. So, the quickest way Söderling can win the game is simply to win the next two points, and his chances of doing that are 25/64.

The only other way Söderling might win is if Söderling and Roddick each win one of the next two points, with the chances of that occurring being 30/64. Furthermore, once back at deuce, Söderling will again have the same chance W of winning. That means that the odds W of Söderling winning from deuce must be a solution to the equation

 

W = 25/64 + (30/64)W

 

We can now easily solve this equation to give W = 25/34. That is, we estimate that Söderling has about a 74% chance of winning from deuce. 

Since 26% of games go to deuce, that will result in another 0.74 x 0.26 = 19% of games that Söderling will win. In total then, Söderling should win 78% of his service games, remarkably close to our original estimate. 

The mathematics seems to check out very well, and we know that the French Open players were very concerned. So, we’ve shot them off a message, assuring them they should continue playing just as they are.

 

Puzzle to ponder: Suppose it is deuce in a certain game, and suppose Roger Federer is serving with a 2/3 chance of winning each point. What are the chances that Federer will win the game?

 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator. 

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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