# Monkey Puzzle

What is the probability that all three monkeys left the cafe wearing the wrong hats?

## Problem

Three monkeys met for tea in their favourite cafe, taking off their hats as they arrived. When they left, they each put on one of the hats at random.

What is the probability that they all left wearing the wrong hat?

## Student Solutions

Let the hats of monkeys A, B and C be a, b and c respectively.

This table shows the ways in which the monkeys can select hats.

A | B | C |
---|---|---|

a | b | c |

a | c | b |

b | a | c |

b | c | a |

c | a | b |

c | b | a |

In only two of the six ways do none of the monkeys have the same hat as when they arrived, hence the required probability is

$$\frac{2}{6}=\frac{1}{3}$$