There are two envelopes in front of you, and you know that one of them has ten times more money than the other. You pick randomly one envelope, but before taking it home, you are given the option to switch, and actually take the other envelope. Should you switch?

That was (one version of) the infamous two-envelope paradox. It's a paradox, because there is a seemingly convincing argument for why switching is pointless (the two envelopes are symmetrical), but also a seemingly convincing mathematical argument for why switching increases your expected winning. This paradox has baffled some of the greatest minds, including the king of recreational mathematics, Martin Gardner.

Unlike most other famous probability paradoxes (e.g., the Monty Hall problem, the "boy or girl" paradox, and Simpson's paradox), the two-envelope problem has a relatively intricate solution. At its heart, the solution is related to the limitations of the notion of expected value, and to an often overlooked caveat regarding conditioning and the law of total expectation (aka the tower property). The paradox solution is also related to the mathematics behind conditional convergence of series, and to Riemann's rearrangement theorem (aka Riemann series theorem).

Created by Yuval Nov for the 2021 "Summer of Math Exposition" (SoME1) competition, hosted by the one and only 3Blue1Brown (Grant Sanderson).


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