Two envelopes problem
brain teaser, puzzle, or paradox in logic, probability, and recreational mathematics
The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory and for the Bayesian interpretation of probability theory. It is a variant of an older problem known as the necktie paradox.
The problem is typically introduced by formulating a hypothetical challenge like the following example:
Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch?
Since the situation is symmetric, it seems obvious that there is no point in switching envelopes. On the other hand, a simple calculation using expected values suggests the opposite conclusion, that it is always beneficial to swap envelopes, since the person stands to gain twice as much money if they switch, while the only risk is halving what they currently have.
“Two envelopes problem” enters the record as brain teaser, puzzle, or paradox in logic, probability, and recreational mathematics. Crown Archives preserves that source wording while asking what envelopes, problem and brain can confirm, complicate or overturn.
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Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Sep 15, 2026. The linked authority identifier is Q180191. None of the 0 selected statements returned an explicit reference.
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This entry incorporates text from “Two envelopes problem” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.