You entered in the Petersburg casino. Your entrance fee is 20 dollars
in each game. In one game,
a coin is thrown repetively until it shows no more head.
You get 2n-20 Dollars, if n times Head appeared, the bank
gets in each game 20-2n dollars.
You will experience that with this entrance fee,
you are loosing 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 expectation value for your win. Whatever entrance
fee you would pay, you would win. The paradox is that this disagrees
with your experience to loose.