In the game of squash the serve passes from one player to another only when the serving player loses a rally. A player wins a point when, and only when, they win a rally on their serve.

Usually the winner is the first player to reach 9 points, but if the score becomes 8-all then the game can be played to either 9 or 10 points: the person who first reached 8 points makes this decision.

Suppose that the score in a game is 8-all and you reached 8 points first and you have a probability of $p$ of winning any particular rally. Under which circumstances is a 9 point game a good idea?

NOTES AND BACKGROUND

Sportspeople often have very clear strategies in their minds when playing different opponents and sometimes make shot decisions based on their chances of winning points in different circumstances: sometimes it is best to 'play it safe' and on other occasions more risky play is called for.

This problem is based on the Traditional International rules of squash, taken fromhttp://www.squashgame.info/squashlibrary/2