Gambling on coin flipping should sound familiar to everyone. In each round of the game, the gambler can make a bet on whether a coin flip will show heads. If it turns out heads, the gambler will win the same amount they bet; otherwise, they will lose this amount. The game continues until the gambler loses (ends up with nothing) or wins (wins more than 100 dollars, let's say). Let's say the coin is unfair and it lands on heads 40% of the time. In order to maximize the chance of winning, how much should the gambler bet based on their current capital in each round? This will definitely be an interesting problem to solve.
If the coin lands on heads more than 50% of the time, there is nothing to discuss. The gambler can just keep betting one dollar each round and should win the game most of the time. If it is a fair coin, the gambler...
