The Monty Hall Explanation

This is not a proof at all, but it seems to be closer to the way of reasoning for many people who at first think that it doesn't matter whether you switch or not:

Imagine there were not three doors, but 100. You choose one of them, and the host opens 98 others with goats behind them, leaving only "your" and one other door.

Would you not think that somehow there's some "information" in the one door spared by the host, some information that was not in your first, random choice? Would't you somehow feel compelled to switch to this other door?

The somewhat more theoretical explanation - maybe even proof - along these lines is:

Your choice splits the doors in two sets. Set A contains the door you selected, and the probability that this is the sports car is 1/3 (or 1/100). The set B contains all remaining doors, and the probability that the winning door is somewhere in there is 2/3 (or 99/100). By removing one (or 98) doors, which all have the success probability of zero because there's a goat behind them, from set B, only one door remains in B, but the overall probability for success in set B is still 2/3 (or 99/100).

If that is not sufficient for you, there's still

Back to the problem descripition

  Frederik Ramm, 2003-02-08