Suppose the chance of winning and the chance of losing a game are both 0.5. You have $A at first, and the bet per game is $1. You stop playing when either you lose all your money (bad outcome) or when you reach $B, where B>A (good outcome). Then the chance of having the good outcome is ##\frac{A}{B}##.

$B is the amount of money you aim to get before you stop playing. Clearly, it doesn't make sense to have an aim of more than $2A because then it's more likely that you will lose all your money instead. So the best strategy is to stop at small wins.

However, once you reach $B, the probabilities get updated, just like the probability of getting another head from a coin toss is still 0.5 after getting 9 heads in a row. So you continue playing, telling yourself you would stop at small wins. But every time you reach your goal, the probabilities get updated, and you would always continue playing.

And if you always continue playing, it means you will go beyond $2A. But this we already showed earlier is not a wise choice.

So should you stop at small wins or not?

$B is the amount of money you aim to get before you stop playing. Clearly, it doesn't make sense to have an aim of more than $2A because then it's more likely that you will lose all your money instead. So the best strategy is to stop at small wins.

However, once you reach $B, the probabilities get updated, just like the probability of getting another head from a coin toss is still 0.5 after getting 9 heads in a row. So you continue playing, telling yourself you would stop at small wins. But every time you reach your goal, the probabilities get updated, and you would always continue playing.

And if you always continue playing, it means you will go beyond $2A. But this we already showed earlier is not a wise choice.

So should you stop at small wins or not?

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