You enter the Petersburg casino. In each game, your entrance fee is $20.
During such a game, a coin is thrown repeatedly until it stops showing "head".
You win 2n-20 dollars, if n times "head" appears. The bank
makes 20-2n dollars.
You will experience that with this entrance fee of 20 dollars
you are losing money. In reality, it is a mathematical fact
that you are expected to win: you win with probability
2-n the amount of 2n dollars. Summing over all
n gives an infinite win expectation. Whatever entrance
fee you would pay, you would be still be in an advantage. The paradox is
that this mathematical finding disagrees with your experience.