A well-known roulette trick, to surely make a profit, is the following: you stake continuously at one color, for example red, double the bet if you lose, and stop as soon as you win. A well-known roulette trick, to surely make a profit, is the following: you stake continuously at one color, for example red, double the bet if you lose, and stop as soon as you win. Because you get twice your bet back if you win, and the ball will once fall on red, you know that you will gain your original bet as profit (you must, however, have an infinite amount of money to be able to double your bet everytime when necessary). The expected value for your profit is therefore equal to your original bet.

But assume that there is a maximum stake for the roulette, which means that you can only stake n consecutive times with this trick.

What is the expected value for your profit for this limited roulette?