User:Tlogmer/Monty Hell problem

The Monty Hell problem is a name for a paradox in probability theory involving infinite sequences of actions. As described in a post in rec.puzzles, the problem consists of choosing between two alternative strategies for banking your money while spending an eternity confined in Hell. The assumptions of the problem are that each day you are paid $10 in $1 bills, but must turn over $1 each day to the Devil to pay for the heat. You are not allowed to handle your money yourself, but instead must choose one of two bankers:

• Monty, who puts each day's bills in a large sack, then chooses one of the bills from the sack uniformly at random (including bills from previous days) to give to the Devil.

• Marilyn, who carefully removes one bill from the stack of ten bills to give to the Devil, and then places the remaining nine bills in the stack.

The goal is to maximize your wealth at the end of your eternal confinement, which occurs on a hypothetical day $\aleph_0$.

Solution

The generally accepted solution is that it is better to bank with Marilyn than Monty. With Marilyn, no bill that ever enters the sack ever comes out again, leaving you with infinite wealth upon your eventual release. With Monty, it can be shown that the probability that any particular bill stays in the sack forever is $0$, which implies that the probability that there are any bills at all left in the sack when you leave Hell is also $0$.

Historical notes

The Monty Hell problem is named after the unrelated Monty Hall problem. In the problem, Marilyn may refer to Marilyn vos Savant, who popularized the Monty Hall problem in her column in Parade magazine.