# Math IA type 2. In this task I will be investigating Probabilities and investigating models based on probabilities in a game of tennis.

Nikhil Lodha

Mathematics Internal Assessment Type 2

Mathematical Modelling

Modelling Probabilities in games of tennis.

IB Math HL

Mr. Hoggatt

2nd June 2009

Introduction

In this task I will be investigating Probabilities and investigating models based on probabilities in a game of tennis. I will look to start with a relatively easy and simplistic models where Adam and Ben play each other in Club practice and have a set number of point that they will play. I will then look to find an expected value for the number of points that Adam wins. For this expected value I will calculate a standard deviation to see how much does a randomly selected point vary from the mean. I will then look at Non Extended play games where a maximum of 7 points can be played. I will show that there are 70 ways in which the game can be played. I will do this with the help of the binomial probability distribution formula.

I will also calculate the odds of Adam winning the game and then look to generalize my model so that it does not only apply to only Adam and Ben but to any player.

After making generalized model, I will look at extended games where in theory games could go on forever. Here I will look to use the sum of an infinite geometric series to come up with an appropriate model. I will then use that model to find the odds of Adam winning extended games and then I will look to generalize this model too.

I will also test the model for different values of point winning probabilities and find out the odds for each of them, I will then look for patterns in the values for odds.

Finally, I will evaluate the benefits and limitations models such as these.

Part 1

The ratio of the points won by Adam and Ben are 2:1 respectively. Therefore Adam wins twice as many points as Ben does. Therefore Adam wins of the points and subsequently Ben wins of the points.

The distribution of X, the number of points won by Adam would be derived by using the binomial probability function and substituting variables and constants to arrive at an appropriate model for the distribution of X, the number of points won by Adam.

The distribution chosen is the binomial probability distribution because in the case of the games of tennis,

• There is a repetition of a number of independent trials in which there are two possible results, success [the event occurs ex. Adam wins the point] or failure [the event does not occur ex. Adam does not win the point].
• The probability of success p, is a constant for all trials.
• The probability of failure q is a constant for all trials.

Here p can substituted by the  since that is probability of Adam winning each point and q can be substituted with  since.

Therefore,  represents the binomial coefficient and it can also be written as

Now, n is substituted with 10 because it’s the number of points which are played between Adam and Ben and we need to find how many points each player wins so r is substituted with A as a variable where it represents the number of points which are won by Adam and therefore  would represent the number of points won by Ben.

The possible limitations to this value might be that although, Ben and Adam know that Adam wins twice as many points as Ben does, still conditions in practice may vary and a change in these conditions such as weather change, injury, illness etc. might lead to change in the probability of the points won by each player. The mathematical model stated above, does not take into the above mentioned factors such as injury or illness [such as dehydration] and this is a limitation because it makes the model unrealistic and merely theoretical. Therefore although it can be used relatively accurately to predict the most probable outcome, still it is not set in stone.

Also unless, the total numbers of points played are multiples of three, the last 2 points or any 2 points will have to be distributed such that for example Adam wins 1.333 of the 2 points and Ben wins 0.667 of the 2 points. This is not practically possible and a limitation to the model.

I do not have concerns about its validity other than the fact that the point winning probability will not be constant for all trials.

Now the binomial distribution model which has been developed above must be used to find the distribution of the points.

The probability of Adam winning ‘A’ number of points can be found by substituting ‘A’ with the number of points that Adam Wins and after solving, the answer will be Adams probability of winning the given number of points that were substituted into the equation.

Data Table 1

Data Table 2

I can also check for reasonability of these values because I know the fact that they must add up to 1.

Therefore I will find the sum of the values of the distributions.

Therefore the values ...