Monty Hell problem

:A different article treats the Monty Hall problem.

The paradox

Let us start with the obvious explanation why it doesn't make any difference which banker you choose: after t days, both Monty and Marilyn have 9t dollars. Since these quantities both grow without limit, either will give you infinitely many dollars in the end.

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Unfortunately, there is a less obvious explanation that favors Marilyn. This explanation depends on the assumption that the contents of Monty's sack on day ω is a set-theoretic limit of the contents on the preceding days, where the limit of a sequence of sets A1, A2, ..., is defined to contain some element x only if there is some number N such that x appears in An for each n geq N. Since it can be shown for any individual dollar bill x that the probability that it remains in the sack forever is zero (see proof below) and since there are only countably many bills, with probability 1 every bill is eventually removed from the sack. Under the assumption that your holdings at the end of time are a limit set defined in this way, Monty almost always leaves you with nothing, and you are better off with Marilyn. This is true even though at any finite time Monty and Marilyn have the same number of dollar bills in their sacks.

Related Topics:
Set-theoretic limit - Countably

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The paradox is the apparent contradiction between these two answers. The paradox is particularly painful because the obvious explanation requires very little mathematics, making the second, non-intuitive answer look suspiciously complex.

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