Snow

A fair coin is tossed. In the first round, you bet on 1$. If it turns out head, then you bet on 1$ for the next round, if it turns out tail, then next round you double your previous bet. In this way, if you have a successive loss, but after some time, you win the round, then you can get all your loss back, plus some little extra. Sounds good? I’ll simulate 10,000 rounds and show you why it’s a bad idea.

1 stands for win, -1 stands for loss

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Imagine there is an urn with n white balls and m black balls. Every time, you draw a ball from the urn and put it back with k more balls of the same color. Repeat the process. The fraction of the white (black) balls in the urn is a martingale. (The expected color in the next draw is the same as the current expected color) Mathematical details coming soon...

It is unnecessary to create the initial urn with a random sampling method, but it’s a little fun thing to do anyway.

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