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.