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Win or Lose?

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?

Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

What's the Point of Squash?

Age 14 to 18
Challenge Level

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?


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