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'What's the Point of Squash?' printed from https://nrich.maths.org/
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 from
http://www.squashgame.info/squashlibrary/2