Jordan Ellenberg in Slate:
Here’s how to make money flipping a coin. Bet 100 bucks on heads. If you win, you walk away $100 richer. If you lose, no problem; on the next flip, bet $200 on heads, and if you win this time, take your $100 profit and quit. If you lose, you’re down $300 on the day; so you double down again and bet $400. The coin can’t come up tails forever! Eventually, you’ve got to win your $100 back.
This doubling game, sometimes called “the martingale,” offers something for nothing—certain profits, with no risk. You can see why it’s so appealing to gamblers. But five more minutes of thought reveals that the martingale can lead to disaster. The coin will come up heads eventually—but “eventually” might be too late. Most of the time, one of the first few flips will land heads and you’ll come out on top. But suppose you get 11 tails in a row. Just like that, you’re out $204,700.* The next step is to bet $204,800—if you’ve got it. If you’re out of cash, the game is over, and you’re going home 200 grand lighter.