# Modelling Probabilities in Tennis. In this investigation I shall examine the possibilities for modelling the probabilities in tennis matches of varying complexity.

Daniel Bregman                14/07/2011

Modelling Probabilities in Tennis

Introduction:

In this investigation I shall examine the possibilities for modelling the probabilities in tennis matches of varying complexity. I will begin by assuming a very simple game of ten points, played by players of a consistent known strength. I will then expand this into games following the real-life rules of tennis more closely, to see how these affect the probabilities involved. I will examine how any results may be generalized to players of different strengths, and how these strengths affect the odds of such players over matches.

A simple model:

Let us assume two players Adam and Ben (who will be referred to as A and B), with fixed probability of scoring a given point against each other. Let A win of points, and B win of points.

We can begin by simulating a 10-point game. As the probabilities are consistent, and there are only two possibilities for each point, the game is a series of Bernoulli trials and can be modelled with a binomial distribution. Let X denote the number of points scored by A. We can now state that:

and therefore that:

.

We can hence easily calculate the probability distribution for all possibly values of X:

This can be presented as a histogram:

This shows that the modal score will be 7, with the highest individual probability. Based on the binomial distribution, we can also calculate the expected value and standard distribution:

Based on this we can see that most scores fall between 4 and 8, with the mean score being 6.6667.

It is important to remember the assumptions we are making here. In real play it is unlikely that a player’s point probability would remain constant throughout play. We have not taken into account factors such as tiredness, confidence etc. which would seriously affect the game. With these caveats in mind, however, we can use the model to predict what might happen in an actual game.

Short game play:

Using the same players A and B from above, we may imagine a version of the tennis rules in which to win a player must have above four points ...