St. Petersburg Paradox

http://en.wikipedia.org/wiki/St._Petersburg_paradox is a 'paradox' where a player pays some fixed amount (say $20) to enter the game, and it appears that the expected gain from the game is infinite dollars. Therefore, the debate is that the player should pay infinite dollars for the entrance (not $20). The counter argument is that the player argues that i've paid $20, but the heads could show up at the first coin flip itself and so i will be left with a loss. Further, even though i may get a lot of money later the probability of that happening is extremely low -- this diminishing probabilities of riches seems to put off most people - yet - mathematically, the "expected" returns are infinite. This is thus argued to be a case for the rejection of mathematical expectation i.e. our psychology goes against the mathematical (long term and statistical certainty) because in real-life we cannot play infinitely long and we do not have infinite (or large) amounts to bet.

Btw, as a tangential (not at all related to betting and expectations but a side-point about the usage of the word 'average') and a somewhat funny point (though unrelated to the expectation issue), is that one can have a population where almost EVERYONE is above average. e.g. if there are just a few people with 1 leg, 1.99 could be the average number of legs in the population which is <2 for almost everyone. Of course this is kind of obvious that we aren't talking about the median, but ive often seen these two things being confused.