So, there is this Carnie (small hands, smells of cabbage) that devised this new game at the carnival and wants to cheat you out of your money. You go to play this game. The game works as such:
The first round you flip a fair coin. If it comes up heads, you get paid $2. However, if it lands tails, you are allowed to go to the next round.
The second round works the same as the first. If it comes up heads, you get paid *$4*. But if it lands tails, you go on to the next.
The third round pays $8 dollars and so on.
If you were allowed to play this game as many time as you wanted, what is the maximum you would pay to start the game? (What is the average payout?)
