I thought about this outta no where. I want to make a casino game where the odds are against the house, but the catch is that the game rules force the player to play until the odds change against him. For example, make the odds 80% against the house (player has 80% chance of winning). If the player puts down a dollar at first and each time he wins, the casino pays him another dollar (doesn't double his money). He is forced to play a certain number of times, but if he loses he loses his initial dollar. Here's how I thought about it (from the player's POV): amount of money won = number of times the game was played = X probability of winning each time = P amount lost = $1 Number of times the game has to be played in order to break even: (amount of money won)*(probability of winning) - (amount lost)*(probability of losing) = 0 X*P^X - ($1)*(1- P^X) = 0 which simplifies to: (P^X)*(X+1) - 1 = 0 All the casino has to do to increase the odds in its favor is to force the player to play at least 1 more time than X (which depends on P). Anyone see any problems with my logic or math? PS. I initially thought this problem would be a lot simpler than this .