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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?

In a Box

Age 14 to 16
Challenge Level

Chris and Jo decide to play a game.

They put some red and some blue ribbons in a box.

They each pick a ribbon from the box without looking (and without replacing them).

Jo wins if the two ribbons are the same colour and Chris wins if the two ribbons are a different colour.


How many ribbons of each colour would you need in the box to make it a fair game?

Is there more than one way to make a fair game?


This problem is based on one offered by Doug Williams at the 2003 ATM conference in Bath UK. See also

Click here for a poster of this problem.