Friday, July 24, 2009

the full Monty

There is one crucial point that I didn't make clear in the last post about the Monty Hall problem. If the probability that the other unopened door is the door with the prize changes after the host's action, information must have been received somehow. It comes from the fact that Monty is a "knowledgeable host"---he knows which door hides the prize, and he always opens a door that reveals a goat.

This contingency is made explicit by considering other potential host behaviors as described in the Wikipedia entry:

• Monty from Hell
• Angelic Monty
• Ignorant Monty
• Monty only offers sometimes

One good way to see that the result is correct is to extend the problem to a deck of cards. Suppose you are to choose among 52 cards, hoping to get the Ace of Spades.

You choose one card, which remains hidden, and now I turn over 50 cards, none of which turns out to be the Ace of Spades. It is pretty clear the probability that your first choice was correct is 1 in 52 and now the odds for the one card remaining are obviously much improved.

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