Monty Hell problem: Encyclopedia II - Monty Hell problem - Solution

Monty Hell problem - Solution

The solution to the paradox is to observe that the two contradictory arguments are taking different limits. The obvious answer takes the limit of the number of bills in the sack; the less obvious answer takes the limit of the set of bills in the sack first, and counts them up afterwards. It is surprising, but not inconsistent, that these two different computations yield different answers.

Which answer is correct? This depends on how you interpret the problem. If you concentrate on the fate of individual bills, and are comfortable with set-theoretic limits, you are likely to take the set-theoretic limit first and conclude that Monty leaves you with nothing. This leaves you in the embarrassing situation of having to explain how you could have had increasing wealth forever but suddenly lose it all at the end, and the assurance that this outcome is consistent with modern mathematics may not make you feel much better. If you treat the individual bills as mere tokens representing your total wealth, you are likely to prefer the numerical limit and conclude that Monty leaves you with infinite wealth. But now you are in the still more embarrassing situation of having to explain how you came to hold a sack that contains infinitely many bills, every single one of which you previously lost forever.

The rec.puzzles discussion ultimately favored the zero-wealth outcome, but the issue continues to be contentious. For further discussion, see Talk:Monty Hell problem.

Adapted from the Wikipedia article "Solution", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki