### Rain or Shine

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

### Squash

If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

### Snooker

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

# Dicey Dice

##### Stage: 5 Challenge Level:

This problem follows naturally from X-Dice, although it may be attempted independently.

A company wishes to produce a set of three different 6-sided dice coloured Apple Green, Bright Pink and Cool Grey, called $A, B$ and $C$ respectively.

They are to be made with the following properties:

1. The faces are to be numbered using only whole numbers 1 to 6.
2. Some of the numbers 1 to 6 can be left out or repeated as desired on each dice.
3. Apple Green is expected to beat Bright Pink on a single roll
4. Bright Pink is expected to beat Cool Grey on a single roll.
5. Cool Grey is expected to beat Apple Green on a single roll.

Invent a set of such dice.

What is the probability that each of your dice wins or loses over each of the other dice?

Is it possible to create a totally fair set of such dice with $P(A> B) = P(B> C) = P(C> A)$?

NOTES AND BACKGROUND

These dice exhibit a property called non-transitivity. Other examples of non-transitivity are found in voting systems. You can read about this in our article Transitivity. You might also wish to try the related problem A Dicey Paradox.