A coin game multiplies the stakes by 5 after every head and ends on the first tail. Every flip has positive expected value. When do you stop?
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You play one round of a game. Pay $1 to enter. Flip a coin: heads adds $5 to your pot; tails loses $1 and ends the game. After a head you may flip again with stakes multiplied by 5 (heads adds $25, tails loses $5 and ends the game), and so on, each time multiplying stakes by 5, until you lose or choose to stop. What strategy gives the highest expected winnings, and when would you actually stop?
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