Probability · experts disagree
The two-envelope problem
Traced to Maurice Kraitchik's 1943 necktie puzzle; the envelope form was spread by Barry Nalebuff in 1989 and Martin Gardner in 1982.
Someone puts money in two envelopes. One holds twice as much as the other. You do not know either amount. You pick an envelope at random, open it, and find 100 dollars. You are now offered the other envelope in exchange.
You reason as follows. The other envelope holds either 200 dollars or 50 dollars. Each seems as likely as the other, since you chose at random. So the swap is worth half of 200 plus half of 50, which is 125 dollars, more than the 100 you hold. And the same argument would have told you to swap whatever amount you had found, which means you should have swapped before opening. Should you take the other envelope?