# Modelling Probabilities in Games of Tennis

Math HL Portfolio Type II:

Modelling Probabilities in Games of Tennis

May 2009

In this portfolio we will look at the probability involved in playing tennis.  Our calculations will be based on the estimated probability a player has of scoring a point.  We will develop models for different kinds of tennis games and use Excel to explore up to what extent we can exploit the two probabilities with which we start.  Furthermore, we will differentiate between probability and odds, comparing them and analyzing how they can affect the way we look at the same numbers.  In my conclusion I will mention the possibility of involving other kinds of distribution in this portfolio, such as Poisson.

Part 1: Club Practice.

1. Games to 10 points.

a) Since we know that Adam wins about twice as many points as Ben does, we can say that the probability of Adam winning a point is, and the probability of Ben winning a point is. So, given that P(A) is the probability of Adam winning a point and P(B) is the probability of Ben winning a point, we have that:

P(A)=

P(B)=

Clearly, this is a binomial distribution.  Hence, we will use the formula, where n is the total number of trials, x the number of successes, p the probability of success and q the probability of failure.  Because we want the variable x to represent the number of points won my Adam, we will substitute P(A) for p and P(B) for q.  Regardless of what is considered a success or a failure, n will be 10.  Following these guidelines our model will be the following:

Where x is the number of points won by Adam.

The only concerns I have about the validity of such a model come from the assumptions made in the question. The words “two players have played against each other often enough to know […]” make me wonder how many times is often enough. Theoretically, if we want to develop a truly pure mathematical formula, both players would have to play against each other an infinite number of times.  This would eliminate any impurity brought about by an external influence. However, this is obviously impossible, so “often enough” will be considered to be a sufficient amount of times to include chance and possible alterations into P(A) and P(B). Possible alterations could be the conditions the players play under.  Adam could play a lot worst when the temperature increases, whereas Ben might play better on hard courts than on grass courts. Chance simply represents either Ben or Adam having a good day. These uncertainties render a mathematical formula inefficient, as there is no way of including every single external factor into the equation.  Therefore, we will work under the assumption that the probabilities include any possible alterations and the frequency with which they might happen.

b) With the model developed in 1.a) and through the use of Excel we can find all the probabilities Adam has of scoring X points with a spreadsheet like the following:

In order to calculate each probability Adam has of winning X points we insert the following command into Excel: =(COMBINAT(10,A)*(2/3)^(A)*(1/3)^(10-A), where the column A stands for X, the number of points won by Adam.  When we graph the data we get a histogram like the one on the next page:

c) To find the expected value and standard deviation we must use the following equations:

We have that n=10 and p= , so

This information tells us that Adam usually wins 7 points, although sometimes he might score a couple of points below or above the mean.  It is very unlikely for him to get more than 8 points or less than 4.

Part 2: Non-extended play games.

2. Different ways in which a game might be played.

The first thing we must do if we want to know all the different ways a game can be played is find ...