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 standard deviation of this binomial random variable has the formula

Where n is the number of trials which would be 10 here, since there are 10 points played. p would be the probability of success and q the probability of failure and they would be

Therefore:

Therefore a randomly selected point varies from the mean by 1.491points and the mean is 6.667.  Thus normally Adam will win  points which means that Adam makes between 5.176 to 8.158points and therefore we can conclude that Adam normally wins the practice games as he almost certainty gets more than half the points, which is more than 5 points.

Part 2

Now I will look at Non extended play games where to win a game, the player must win with at least 4 points and by at least 2 points, but to save court time, no game is allowed to go beyind 7 points. This means that if deuce is called and each player has 3 points then the next point determines the winner.

I want to show that the number of different ways in which the game can be played is 70 different ways and I will use primarily the binomial coefficient and lucid logic to show this.

The number of possibilities for the points played can be found by using the binomial coefficient formula which is   .

Therefore in order to find the number of possible games played, I will substitute n with the number of points up for contention [for the sake of calculation, this number is not always the total number of points played because the last point must always be won by the player whose probability of winning is being calculated therefore the n will always be the total points played subtracted by 1.]

Also, players and fans can have a realistic idea of the abilities and expectations. For example

There are however some limitations to the model, the main reason why they are not entirely realistic is that the point probabilities are not always constant in reality because of factors such as changing weather conditions, possible injuries and illnesses.

Also, it has been stated that the ‘2 players, Adam and Ben have played each other often enough to know that Adam wins about twice as many points as Ben does’. Even if we assume that they have practiced against each other and played non-extended games against each other enough to know that their point winning ratio is 2:1, still for extended games such with the possibility of games going on indefinitely, we cannot assume that the their point winning ratio will hold as time goes on, simply because they Adam and Ben could not have played indefinitely, in fact they may not have play each other beyond a couple of deuces. Therefore the fact that the point winning ratio is constant in the model but realistically is changing is a limitation to such probability models.  

Another limitation or rather downside to such a model would be that the element of surprise of anticipation of the fans would not exist as the expected outcome would tell us the most likely results and watching the game would no longer be enjoyable, the players would be like robots merely doing their job and playing as math dictates!