Solution to: Faites Vos Jeux
Let the original bet be equal to b. The probability that you lose a round is 19/37. The probability that you lose n consecutive rounds (which means that you must quit with a loss) is (19/37)n. Therefore, the probability that you win once is 1 - (19/37)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 (19/37)n × (-(2n - 1) × b) + (1 - (19/37)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!
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