## Answer

So let's work through this together.

Clearly there are three possibilities for the initial state. Let's call the initial states P1, P2 and P3.

Possibility |
Container 1 |
Container 2 |
Container 3 |

P1 |
$1,000,000 |
$10 |
$10 |

P2 |
$10 |
$1,000,000 |
$10 |

P3 |
$10 |
$10 |
$1,000,000 |

### If we take a look at initial state P1

**If you were to choose container 1...**

The question states that the host will open a different container, and that it will contain $10.

Suppose the host opens container 2. It contains $10.

If you stick with your choice (container 1), you ... WIN

If you switch your choice to container 3, you ...... LOSE

#### Remembering that P1, P2 and P3 are equally likely, we can sum up in this table

P1/P2/P3 |
You choose |
Host opens |
You stick |
You switch |

P1 |
C1 |
C3 |
WIN |
LOSE |

P1 |
C1 |
C2 |
WIN |
LOSE |

P1 |
C2 |
C3 |
LOSE |
WIN |

P1 |
C3 |
C2 |
LOSE |
WIN |

#### So... It's a 50/50 chance whether you should stick or change, right ?

Well, no. Not really.

Your choices, C1, C2 or C3, are all equally likely.

So it should really look like this :

P1/P2/P3 |
You choose |
Host opens |
You stick |
You switch |

P1 |
C1 |
C3 or C2 |
WIN |
LOSE |

P1 |
C2 |
C3 |
LOSE |
WIN |

P1 |
C3 |
C2 |
LOSE |
WIN |

### Summing it all up.

#### Now each case P1, P2 and P3 are intuitively the same, and indeed they are.

P1/P2/P3 |
You choose |
Host opens |
You stick |
You switch |

P1 |
C1 |
C3 or C2 |
WIN |
LOSE |

P1 |
C2 |
C3 |
LOSE |
WIN |

P1 |
C3 |
C2 |
LOSE |
WIN |

P2 |
C1 |
C3 |
LOSE |
WIN |

P2 |
C2 |
C1 or C3 |
WIN |
LOSE |

P2 |
C3 |
C1 |
LOSE |
WIN |

P3 |
C1 |
C3 |
LOSE |
WIN |

P3 |
C2 |
C1 |
LOSE |
WIN |

P3 |
C3 |
C1 or C2 |
WIN |
LOSE |

So, if we stick, we have 3 wins out of 9.

If we switch, we have 6 wins out of 9

In 2/3 of all cases, it's better to switch. In other words, it makes sense to switch!