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!

Back to the puzzle