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Doubling up until you lose

expected-valuegamblingmartingalerisk

You start with 1 dollar and a fair coin. You bet your entire bankroll on heads. If you win, your bankroll doubles and you bet the whole thing again. You plan to stop after n consecutive wins, or when you first lose.

  1. What is the expected value of your final bankroll?
  2. What is the probability that you end with more than you started with?
  3. What does this say about using expected value alone to judge a gamble?
Show a hint

Each bet is fair, so the expected value after each bet equals the value before it. Then look at what actually happens in almost every run.

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The Coupon-Collector ProblemCraps: Working Out the House EdgeThe Dice Game Pig and Optimal StoppingWhy the Doubling Betting System FailsExpected Distance Between Two Random Points

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