### 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?

# Non-transitive Dice

##### Stage: 3 and 4 Challenge Level:

Here are three dice that are used to play a game for two players:

The red die has the numbers {1, 1, 6, 6, 8, 8}

The green die has the numbers {2, 2, 4, 4, 9, 9}

The blue die has the numbers {3, 3, 5, 5, 7, 7}

Each player chooses a different die.
They roll their dice.
The winner is the person whose die shows the bigger number.

Alison and Charlie are playing the game. Charlie wants to go first so Alison lets him.

Was that such a good idea?

Can you advise Alison on which die to choose once she knows which die Charlie has selected?

Notes and background

Dice of this sort are known as non-transitive dice. You can read more about transitivity in this article or have a go at creating your own in Dicey Dice. This spreadsheet might be useful if you want to create your own.

After you've had a go at the problem, you may be interested to read more about dice in the Plus articles Curious dice and Let 'em roll.

You may be familiar with another non-transitive game known as 'rock, paper, scissors'.