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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