Question

You and a friend play the following game: You pay your friend $3 each turn and then flip a fair coin. It it’s tails, your friend pays you $(2^n), where n is the number of times you’ve flipped the coin, and the game ends. If it’s heads, you have the choice of stopping and continuing. If you have m dollars to start with, and you play the game either until you win or until you have no money left, what will you win on the average?

Ah the good old St Petersburg Paradox.

Well, if you and your friend each have a infinite amount of money, then you can expect to win and infinite amount, on average.

Google St Petersburg Paradox for more info.