Solution to: Faites Vos Jeux
Let the original bet be equal to b. The probability that you lose a round is 18/37. The probability that you lose n consecutive rounds (which means that you must quit with a loss) is (18/37)n. Therefore, the probability that you win once is 1-(18/37)n. If you lose n consecutive rounds, your total bet (and loss) equals (2n-1) × b. If you win once, your profit equals b. So, the expected value for your profit is (18/37)n × ( -(2n-1) × b ) + ( 1-(18/37)n ) × b = ( 1-(38/37)n ) × b.
Note that this expected value is negative for all values of n greater than 0. You cannot expect a profit, but a loss!