Should you always swap the envelope?
You're holding one of two sealed envelopes and know one contains exactly twice as much money as the other. Before opening it, you reason: my envelope has some amount X, so the other has either X/2 or 2X with equal chance, meaning switching gains 25% on average. But the person holding the other envelope can make the identical argument. Both can't profit by swapping unopened envelopes. Where does the reasoning break?
Reveal the answer
The flaw is treating "the other envelope holds X/2 or 2X with equal probability" as true for every possible X, an assumption that can't hold under any genuine probability distribution over amounts. No single fix commands universal agreement; it remains debated in probability theory. Its ancestor is Maurice Kraitchik's 1930 necktie paradox; Martin Gardner reframed it with envelopes, and Barry Nalebuff formalized the now-standard version in 1989.