http://en.wikipedia.org/wiki/Two_envelope_problem
You are given two envelopes to choose from, and are told that one has twice the amount of money as in the other. You pick one and open it, finding $20. The host asks you if you would like to switch. It is apparent from the random nature of the initial choice that switching would not change the payoff.
The only possible amounts in the other envelope are $10 and $40. I became tempted to assign 2/3 probability to $10 and 1/3 probability to $40, so that the expected value came out to $20. This is wrong, however, because the host could announce that he would pay a square of whatever amount of money I ended up with ($100 for $10, $400 for $20, $1600 for $40) and still switching would not matter, yet those probabilities (2/3, 1/3) would prescribe switching.
This is apparently a state of ignorance in which the possibilities are finite and known, and yet probabilities cannot be assigned to them. This disturbs me.
HT: Lee Hsien-Yang’s New York talk last Friday