You have two envelopes—one contains twice as much as the other. Should you switch? Always? Never? The math says... both!
You're given two identical envelopes. One contains twice as much money as the other, but you don't know which is which.
You pick one envelope and peek inside: it contains $100. Now you're offered a chance to switch.
Should you? The other envelope has either $50 or $200. If you calculate the expected value of switching: 0.5 × $50 + 0.5 × $200 = $125. That's more than $100—switch!
But wait... you could make the exact same argument about any amount. And after switching, the same logic says switch back! What's going on?
You open your envelope and see X dollars. The other envelope contains either X/2 or 2X with equal probability:
Since 1.25X > X, you should switch! But this reasoning is flawed.
Let's say the smaller amount is S. The envelopes contain S and 2S. You have one of them with 50% probability each:
Both expectations are equal! There's no advantage to switching.
The fallacy uses "X" to mean two different things: in the X/2 case, X is the larger amount; in the 2X case, X is the smaller amount. You can't use the same variable for both!
The probability that you hold the smaller envelope is 50%, but that's BEFORE you see any amount. Once you see a specific value, the probabilities may not still be 50-50 (they depend on how the amounts were chosen).
For the faulty argument to work for ALL possible amounts, you'd need an infinite pool of money. No finite setup can make switching always better, because the expectation calculation becomes undefined.
Both envelopes are identical before you choose. Any advantage to switching would mean the other envelope is systematically better—but by symmetry, neither can be systematically better!
See what happens over many trials with different strategies:
The paradox traces back to 1943 when Belgian mathematician Maurice Kraitchik posed a puzzle about two men comparing the worth of their neckties—each thinking they have the advantage in a trade.
The modern "two envelopes" formulation appeared in the late 1980s, introduced by Martin Gardner (1989) and Barry Nalebuff (1988).
It has since become one of the most debated problems in probability theory, with papers still being published offering new perspectives on its resolution.