Go see the section on las vagas math. There is a whole thread on that topic.
The following was confusing me so any help would be greatly appreciated!
A gambling book recommends the following strategy for the game of roulette. It recommends that a gambler bet 1 euro on red. If red appears (with probability 18/38), then the gambler should take his or her profit and quit the game. If the gambler loses his or her bet (which has probability 20/38 of happening), he should make additional 1 euro bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote the gambler's winnings when he quits.
Compute the expectation of X.