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Monty Hell problem - The paradox | A Wisdom Archive on Monty Hell problem - The paradox | | Monty Hell problem - The paradox A selection of articles related to Monty Hell problem - The paradox | |
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More material related to Monty Hell Problem can be found here:
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Monty Hell problem, Monty Hell problem - Appendix: Proof that each bill leaves the sack with probability 1, Monty Hell problem - Attacks on the second solution, Monty Hell problem - Everybody dies but that doesn't mean someday no one will be alive, Monty Hell problem - Historical notes, Monty Hell problem - Solution, Monty Hell problem - The paradox, Monty Hell problem - What if the Devil pays you out of his heating fee receipts?, Monty Hell problem - You can't multiply a zero probability by infinitely many elements, Ross-Littlewood Paradox, a problem which is equivalent to this one, but is much more astonishing as it has nothing to do with probability.
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ARTICLES RELATED TO Monty Hell problem - The paradox | | | |  Monty Hell problem - The paradox: Encyclopedia II - Monty Hell problem - The paradoxLet 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.
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 A ...
See also:Monty Hell problem, Monty Hell problem - The paradox, Monty Hell problem - Attacks on the second solution, Monty Hell problem - Everybody dies but that doesn't mean someday no one will be alive, Monty Hell problem - You can't multiply a zero probability by infinitely many elements, Monty Hell problem - What if the Devil pays you out of his heating fee receipts?, Monty Hell problem - Solution, Monty Hell problem - Appendix: Proof that each bill leaves the sack with probability 1, Monty Hell problem - Historical notes Read more here: » Monty Hell problem: Encyclopedia II - Monty Hell problem - The paradox |
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| | | | Monty Hell problem - The paradox: Encyclopedia II - Monty Hell 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.
Monty Hell problem - 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 ...
See also:Monty Hell problem, Monty Hell problem - The paradox, Monty Hell problem - Attacks on the second solution, Monty Hell problem - Everybody dies but that doesn't mean someday no one will be alive, Monty Hell problem - You can't multiply a zero probability by infinitely many elements, Monty Hell problem - What if the Devil pays you out of his heating fee receipts?, Monty Hell problem - Solution, Monty Hell problem - Appendix: Proof that each bill leaves the sack with probability 1, Monty Hell problem - Historical notes Read more here: » Monty Hell problem: Encyclopedia II - Monty Hell problem - Attacks on the second solution |
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| | | | Monty Hell problem - The paradox: Encyclopedia II - Monty Hell problem - Appendix: Proof that each bill leaves the sack with probability 1 Here is a proof that the probability that a bill stays in the sack forever is zero. Let w be the number of bills added each day, and consider some bill that is added on day t0, where the first day is day 1. The probability that it remains in the bag on day n, given that it is present at the end of day n − 1, is
so the prob ...
See also:Monty Hell problem, Monty Hell problem - The paradox, Monty Hell problem - Attacks on the second solution, Monty Hell problem - Everybody dies but that doesn't mean someday no one will be alive, Monty Hell problem - You can't multiply a zero probability by infinitely many elements, Monty Hell problem - What if the Devil pays you out of his heating fee receipts?, Monty Hell problem - Solution, Monty Hell problem - Appendix: Proof that each bill leaves the sack with probability 1, Monty Hell problem - Historical notes Read more here: » Monty Hell problem: Encyclopedia II - Monty Hell problem - Appendix: Proof that each bill leaves the sack with probability 1 |
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| | | | Monty Hell problem - The paradox: Encyclopedia II - 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 ...
See also:Monty Hell problem, Monty Hell problem - The paradox, Monty Hell problem - Attacks on the second solution, Monty Hell problem - Everybody dies but that doesn't mean someday no one will be alive, Monty Hell problem - You can't multiply a zero probability by infinitely many elements, Monty Hell problem - What if the Devil pays you out of his heating fee receipts?, Monty Hell problem - Solution, Monty Hell problem - Appendix: Proof that each bill leaves the sack with probability 1, Monty Hell problem - Historical notes Read more here: » Monty Hell problem: Encyclopedia II - Monty Hell problem - Solution |
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