Monty Hall’s Probability Paradox with Marilyn Von Savant

If a contestant, in a TV show for example, to choose one door out of three doors and only one door has the prize, it is said that the contestant will have 1/3 probability chance of getting the prized door which is essentially same as stating that there is a 2/3 probability of getting the door without the prize.

But once the host of that show opens one of the two doors that contestant didn’t choose and that door didn’t have the prize, the probability changes as the number of unknown doors changes from original 3 to 2 doors. Many people say that now the remaining 2 doors have equally 50-50 chance of having the prize inside and thus don’t matter whether the contestant decides to change from original door of his choice to the single door that is left.

Marilyn Von Savant, however, believes switching from the choice of his original door to the door that is left is better because the latter door carries 66.7% probability of having prize inside now.

Here is how it can be explained in two different ways:

- There was initially 2/3 chance of having prize inside the two remaining doors that contestant didn’t choose. Now that there is only one door left of the two, there is 2/2 chance that the prize is inside this door. The door contestant chose initially had 1/3 chance; now it has 1/2 chance of carrying the prize.
- The probable way the game will be played is limited to 6 different ways and it turns out switching offers 2/3 chance to win as well:

Probable ways the game will be played | Chosen Door | Door 2 | Door 3 | Result |

1 | Prize | None | None | Switch and Loss |

2 | None | Prize | None | Switch and Win |

3 | None | None | Prize | Switch and Win |

4 | Prize | None | None | Stay and Win |

5 | None | Prize | None | Stay and Loss |

6 | None | None | Prize | Stay and Loss |

As this table clearly shows: if you switch, you win 2/3 and lose 1/3; where as if you stay, you win only 1/3 and lose 2/3. This table uses all the probabilities the game will be played and doesn’t employ the premise that the host always knows which door has the prize because some people believe that if the host knows that the prize is not behind the door 3, for example, and therefore he opened it as opposed to opening right away what the contestant chose to go with.