This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
I pick two numbers, randomly, and tell you one of them. You are supposed
to guess whether this is the lower or higher one of the two numbers I
picked. Can you come up with a method of guessing that does better than
picking the response "low" or "high" randomly (i.e. probability to guess
right > .5) ?
Pick any cumulative probability function P(x) such that a > b ==> P(a)
> P(b). Now if the number shown is y, guess "low" with probability
P(y) and "high" with probability 1-P(y). This strategy yields a
probability of > 1/2 of winning since the probability of being correct
is 1/2*( (1-P(a)) + P(b) ) = 1/2 + (P(b)-P(a)), which is > 1/2 by