The envelope paradox is based on the following scenario.

You are, at random, given one of two sealed envelopes that contain money. You do not know the sums in the envelopes but do know that one envelope contains twice the amount in the other. You are given the option to swap your envelope for the other. Should you?

It seems clear by symmetry that there is no benefit in swapping—which is the correct answer—but the following argument to the contrary is perhaps initially convincing. Denote by X the amount in your envelope; the other amount is either 2X or ½X. So you have a half chance of gaining X and a half chance of losing ½X and so an expected gain of ½(X − ½X ) = ¼X>0 and so you should swap.

Different stances can be taken on what is erroneous about this argument. It may be seen as an abuse of notation by letting X denote different amounts in the two different scenarios or equivalently not recognizing the swapping is not independent of the dealing. See also Bertrand’s paradox, Monty Hall problem.