- Level: International Baccalaureate
- Subject: Maths
- Word count: 1624
Modelling Probabilities in Tennis. In this investigation I shall examine the possibilities for modelling the probabilities in tennis matches of varying complexity.
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Introduction
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:
x | P(X=x) |
0 | 0.000017 |
1 | 0.000339 |
2 | 0.003048 |
3 | 0.016258 |
4 | 0.056902 |
5 | 0.136565 |
6 | 0.227608 |
7 | 0.260123 |
8 | 0.195092 |
9 | 0.086708 |
10 | 0.017342 |
This can be presented as a histogram:
This shows that the modal score will be 7, with the highest individual probability.
Middle
0.2963
6
0.2195
7
0.2195
This can be shown as a histogram:
Based on this data we can calculate the probability that A wins the game:
The odds that A wins are therefore:
For more general players C and D with point probabilities c and d, the probability that C wins is given by:
Note that because on one point C and D’s probabilities of scoring are mutually exclusive and exhaustive, we can say that:
In this way the probability of C winning can be written entirely in terms of C’s probability.
Longer game play:
In the above examples the rules ensured that y, the number of points played, could not exceed 7. If instead the rules become that to win, a player must have 4 points and be 2 points ahead of their opponent, we can produce theoretically infinite games. The word ‘deuce’ will be used to refer to a state where the scores are equal and greater than 2, and the word ‘advantage’ will refer to the state after deuce where there is a 1 point difference.
To model this situation, we can consider two possibilities: a game with deuce and a game without deuce. We will discuss the latter first.
If the game is to occur without deuce, it must end at 4-0, 4-1, 4-2, or the reverses of these. Any other score would either be too low to win, or would have involved deuce (e.g.
Conclusion
c | P(Cwins) | odds |
0 | 0.000000 | 0:1 |
0.05 | 0.000092 | 1:10830 |
0.1 | 0.001447 | 1:689.70 |
0.15 | 0.007137 | 1:139.11 |
0.2 | 0.021779 | 1:44.916 |
0.25 | 0.050781 | 1:18.692 |
0.3 | 0.099211 | 1:9.0795 |
0.35 | 0.170355 | 1:4.8701 |
0.4 | 0.264271 | 1:2.7840 |
0.45 | 0.376852 | 1:1.6536 |
0.5 | 0.500000 | 1:1.0000 |
0.55 | 0.623149 | 1.6536:1 |
0.6 | 0.735729 | 2.7840:1 |
0.65 | 0.829645 | 4.8701:1 |
0.7 | 0.900989 | 9.0795:1 |
0.75 | 0.949219 | 18.692:1 |
0.8 | 0.978221 | 44.916:1 |
0.85 | 0.992863 | 139.11:1 |
0.9 | 0.998552 | 689.70:1 |
0.95 | 0.999908 | 10830:1 |
1.0 | 1.000000 | 1:0 |
The relationship can be shown graphically:
This shows that when the point probabilities are close to each other (i.e. near 0.5) a small change can make a large difference in the overall game probabilities, whereas if the point probabilities are far apart a change will not make much difference.
In practical terms, this means (perhaps unsurprisingly) that matches with closely-matched players are the most exciting to watch, because a small change in the player’s relative performance can change the likelihoods dramatically, making the game less predictable.
Conclusions:
These results must be interpreted based on the limitations identified at the start. The results hold for cases with perfectly consistent players, but this is not the case in real life. The changes in performance that would result from tiredness as the game goes on, or from confidence (or the lack thereof) as one player succeeds and the other does not, would significantly change the probabilities in ways that this model cannot predict. Because of this the model would need to be made much more complex if it were to serve as a predictor of real-life situations. However, as a tool to explore the mechanisms at play in a game of tennis, the simple model that we have developed shows clearly how the players’ relative strengths affect the overall result, and also how the rules of play that allow a game to go on longer significantly enhance a strong player’s advantage, so the model is not entirely useless.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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