Monty Hell problem

:A different article treats the Monty Hall problem.

Attacks on the second solution

Because the second solution is so disturbing, many people attempt to resolve the paradox by finding an error in it. This section will describe some of these approaches, and explain why they are not supported by modern-day set theory and probability theory.

Related Topics:
Set theory - Probability theory

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Everybody dies, but that doesn't mean someday no one will be alive

Consider the following variant of the "Monty" process, which eliminates the probabilities: on day 1, element 1 is placed in the sack. On day 2, element 2 is placed in the sack, and in general, on day t, element t is placed in the sack. Starting on day 101, we also remove elements; on day 101, we remove element 1, and in general, on day t+100 we remove element t. Since there are obviously 100 elements exactly in the sack from day 101 on, it would appear that there must continue to be 100 elements in the sack in the limit.

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While it is true that the limit as t goes to infinity of the number of elements in the sack is 100, it is not true that the limit as t goes to infinity of the set of elements in the sack is a set of 100 elements. This is immediate from the definition of a limit of a sequence of sets: there is no element that stays in the sack forever, so the limit is the empty set.

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A similar variant on the "Monty" process would be to put bills numbered 10n-9 through to 10n in the sack on day n, and then to take out the bill numbered n. Each bill will enter and leave the sack at determined times; although the number of bills in the sack will increase without limit over time, no identifiable bill can remain indefinitely in the sack.

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The general rule is that just because every set in a sequence has some property, it doesn't necessarily mean that the limit (if it exists) also has that property. We can see this in the "Marilyn" process as well—the number of bills in Marilyn's sack on each day is finite, but the number of bills in the limit is not.

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This is an astonishing property of set-theoretic limits, and it may surprise the reader to learn that the definition we have been using is in fact the definition universally used in mathematics. The reason this definition is used is that there is no good alternative: if we want to define the limit of the sets of elements of the bag to be some particular 100-element or potentially infinite set, we have to ask which elements are in this limit?

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You can't multiply a zero probability by infinitely many elements

A second objection to the zero-return argument is that it cheats in going from the claim that, for each bill x, the probability that x remains in the bag is zero, to the claim that the probability that there is any bill that remains in the bag is zero. The argument is that, while it might be reasonable to sum up the zero probabilities of finitely many improbable events and conclude that their union also has zero probability, in this case where are summing infinitely many zeroes, and the product of infinity and zero is undefined.

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Though there are axiomatizations of probability theory that do not allow summing over infinite sets of events, the usual set of probability axioms due to Kolmogorov explicitly provides countable additivity, which allows such sums as long as the infinite set is countable. So this particular objection requires---at minimum---adopting a non-standard approach to probabilities.

Related Topics:
Probability axioms - Kolmogorov - Countable

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What if the Devil pays you out of his heating fee receipts?

The second solution assumes that each bill enters Monty's sack only once. But what if your $10 pay comes from the Devil, and he frugally funds some of that pay by recycling your heating fees? In this case each bill may enter the sack infinitely often, and it is not clear what the contents of the sack are in the limit (indeed, depending on how the Devil choose which bills to reuse and when, a limit set may not even exist).

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This evades one version of the paradox, but it does not solve it in general. If the problem is explicitly stated so that the heating fees are not returned (say, because these bills are immediately fed to the infernal furnace), then the paradox returns.

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