Talk:Two envelopes problem

The forgotten fact
If the amount of money is odd, it's better to switch. Since it can't be the X2 enveloppe. If it's even, switching is slightly riskier since maybe one of the envelopes is odd amount of money and you want to avoid it... — Preceding unsigned comment added by 70.48.133.1 (talk) 11:00, 6 November 2017 (UTC)
 * Nice comment. However, as far as we know, there are just slips of paper in the envelopes with the amounts of money written on them. The envelopes need not contain US bank notes. Richard Gill (talk) 07:48, 21 July 2020 (UTC)

New publication coming up
My manuscript on TEP is now posted on arXiv (will appear on Tuesday) and submitted to a journal. Here is a sneak preview. https://www.math.leidenuniv.nl/~gill/tep.pdf Richard Gill (talk) 10:08, 7 March 2020 (UTC)


 * Nice paper, but you might consider removing your reference to Talk:Two_envelopes_problem/Literature as this turned out to be an illegal talk page and will consequently soon be deleted. iNic (talk) 12:18, 7 July 2020 (UTC)


 * That doesn't worry me. Richard Gill (talk) 14:36, 14 July 2020 (UTC)


 * Ok that's good but I'm quite sure it will be deleted from Wikipedia. iNic (talk) 11:43, 15 July 2020 (UTC)

Don't put article content in Talk page
Ref 4 in the current version points to Talk:Two_envelopes_problem/Literature. This makes that Talk sub-page in effect part of the article. This needs resolving: article content should be in the main namespace, not the Talk namespace and citations on Wikipedia shouldn't be pointing to other pages on Wikipedia. The papers in that literature list (the ones that support the statement) need to be moved into the article as citations. MartinPoulter (talk) 18:06, 2 July 2020 (UTC)
 * I changed the citation to a single WP:RS. Rolf H Nelson (talk) 22:57, 5 July 2020 (UTC)
 * Haha so one randomly chosen single paper is now supporting the claim that there is a "voluminous literature"? This is silly beyond belief. iNic (talk) 09:35, 6 July 2020 (UTC)
 * If it's helpful, I added a relevant quote from the paper. Rolf H Nelson (talk) 02:55, 7 July 2020 (UTC)
 * Since material on Wikipedia is never permanently deleted the list of literature is here to stay for ever, and people outside of Wikipedia can refer to its existence. Too bad. The rules of Wikipedia are mutually inconsistent. Wikipedia must disappear and be replaced by an even more complete encyclopedia with rules which are even more difficult to figure out. Naturally, being a compendium of all existing knowledge, it will be able to guide anyone to anywhere on any version of Wikipedia which ever existed in the past. I think we should cherish anomalies like this, not get upset about them. Richard Gill (talk) 14:08, 14 July 2020 (UTC)
 * Anyway, the Wikipedia circular references rule says "An exception is allowed when Wikipedia itself is being discussed in the article, which may cite an article, guideline, discussion, statistic, or other content from Wikipedia (or a sister project) to support a statement about Wikipedia. Wikipedia or the sister project is a primary source in this case, and may be used following the policy for primary sources. Any such use should avoid original research, undue emphasis on Wikipedia's role or views, and inappropriate self-reference. The article text should make it clear the material is sourced from Wikipedia so the reader is aware of the potential bias." It is not clear that this is the only exception which is allowed. New exceptional cases can arise. The rules of Wikipedia need to be adapted to reality, not the other way round. Possibly we should add references to the relevant story by Borges, and possibly add a subsection to the article mentioning the notable fact that the wikipedia article itself *did* become a reliable source in reliable scientific literature. Self-reference is a wonderful source of puzzles! Richard Gill (talk) 14:44, 14 July 2020 (UTC)

Another reference
A correspondent tells me that the following paper is not referenced and seems to them to be the definitive solution: The Two-Envelope Paradox Resolved. Author(s): Timothy J. McGrew, David Shier and Harry S. Silverstein. Source: Analysis, Jan., 1997, Vol. 57, No. 1 (Jan., 1997), pp. 28-33 Published by: Oxford University Press on behalf of The Analysis Committee. Stable URL: http://www.jstor.com/stable/3328431 Richard Gill (talk) 07:50, 21 July 2020 (UTC)


 * There are indeed many solutions and ideas not mentioned in the article. Why? Well there are simply too many of them and many are quite technical and thus not easily explained in an encyclopaedic article. Wikipedia can never be a replacement for the literature itself. An obvious fact that many people need to be reminded of over and over. This paper was, however, referenced indirectly until just recently in Talk:Two_envelopes_problem/Literature. It is the first entry for 1997. But this list is now so kindly removed as a reference by Rolf H Nelson so yes, now this paper is not mentioned at all at Wikipedia. iNic (talk) 11:10, 21 July 2020 (UTC)

List of inconsistencies
I found an interesting list of inconsistencies of the article here iNic (talk) 02:00, 25 November 2020 (UTC)


 * Latest version of that article is dated 2018-01-02. A quick glance lets me doubt its value even when it was current. Paradoctor (talk) 02:28, 25 November 2020 (UTC)

The correct answer by OpenAI
At last we can put an end to this long discussion. The correct answer is really easy.

Question: What is the correct solution to the two envelope problem?

ChatGPT: The two envelope problem is a classic problem in probability theory. The basic premise is that you are given two envelopes, each containing a certain amount of money. One envelope contains twice as much money as the other. The question is, if you are allowed to open one envelope and see its contents, should you switch to the other envelope to maximize your chances of getting the larger amount of money?

The correct solution to this problem is to always switch envelopes. This may seem counterintuitive, but the reason it works is because of the way the problem is set up. Since one envelope contains twice as much money as the other, if you choose an envelope at random, you have a 50% chance of choosing the envelope with the smaller amount of money and a 50% chance of choosing the envelope with the larger amount of money.

If you choose the envelope with the smaller amount of money, and you then switch to the other envelope, you are guaranteed to get the larger amount of money. This is because the other envelope must contain the larger amount of money, since the one you chose contains the smaller amount. Therefore, by switching envelopes, you will always get the larger amount of money.

On the other hand, if you choose the envelope with the larger amount of money, and you then switch to the other envelope, you will only get the smaller amount of money. This is because the other envelope must contain the smaller amount of money, since the one you chose contains the larger amount. Therefore, by switching envelopes, you will only get the smaller amount of money in this case.

Therefore, the correct strategy is to always switch envelopes. This will maximize your chances of getting the larger amount of money.iNic (talk) 11:59, 9 December 2022 (UTC)
 * Hard to believe no one thought of that before! EEng 12:22, 9 December 2022 (UTC)
 * That's no way to make money. Invest in crypto. Tell all your friends! Thincat (talk) 10:37, 10 December 2022 (UTC)
 * Personally I don't think we should have the Talk:Two envelopes problem/Arguments sub-talk page at all—but if any topic belongs there, shouldn't it be this one? Awwright (talk) 18:05, 7 September 2023 (UTC)
 * Yes, a joke post passing on a ridiculous AI-blather "solution" to the problem is what we need here for sure. EEng 19:17, 7 September 2023 (UTC)

Would be useful to put the simple, non-technical explanation first
Although "the person stands to gain twice as much money if they switch, while the only risk is halving what they currently have", what this omits to mention is that they stand to double only the SMALL amount, but risk losing half the LARGE amount, so in fact the gains and losses even out, and the "paradox" evaporates. I think it would be useful to mention this up front, for the benefit of readers who maybe can't cope with the more mathematical explanations. 2A00:23C8:7B0C:9A01:87F:AA06:BD33:A284 (talk) 13:30, 15 April 2024 (UTC)