The discovering mathematics module has made me curious about things in everyday life that involve maths. Squash is a fast game which requires a good level of physical fitness and agility but also a great deal of maths is involved in playing the game.

In a game of squash you can only win a rally on your own serve and the serve only passes from one player to another when the serving player loses a rally. The winner is usually the player first to reach 9 points. However if it is 8 all the game can be played to 9 or 10. The person who is receiving or in other words lost the point decides this. So what should you decide? This is where maths comes into the game?

The answer all comes down to probability which of course is a fundamental mathematics principle.

According to Beardon (2001)

‘Assume players are called A and B. Assume that the probability of A winning a rally is p, whether he serves or not, and whatever the score is. The probability for B is q, where p+q=1

and

The rules of squash say that if the position is reached when the score is (8,8) (we give A’s score first), and B is to serve, then A must choose between the game ending when the first player reaches 9 or when the first player reaches 10. We want to decide what is the best choice for A to make (when p and q are known).

If A chooses 9, then he wins the game with probability φ. Suppose now that A chooses 10; then the sequence of possible scores and their associated probabilities are as follows:

Sequence of scores | Probability | Probability in terms of p,q |
---|---|---|

(8,8)→(9,8)→(10,8) | φ×θ | p3/(1−pq)2 |

(8,8)→(8,9)→(9,9)→(10,9) | μ×φ×θ | p3q/(1−pq)3 |

(8,8)→(9,8)→(9,9)→(10,9) | φ×λ×φ | p4q2/(1−pq)3 |

After simplification ( p+q=1), this is equivalent to p2−3p+1>0.