Did you spot the small inaccuracy in the work given above. The one assumption that can cause a problem?

]]>Is there a notebook?

]]>Huygen? I thought it was Pascal and Fermat.

]]>This is a standard Gambler's ruin problem first expressed by Christian Huygen.

There are 2 possibilities. Maria can go broke or Maria can keep winning indefinitely and not go broke. Obviously Sam can never lose because he can not go broke.

The relevant formula is:

where i is the amount Maria starts with i = 10 bets. n = ∞ which means Maria never goes broke even in an infinite number of plays. p = 2 / 3.

Plugging in we get:

So Maria has a 99.90234375% chance of winning indefinitely and a .09765625% chance of going broke.

It has been known since the time of Huygen that if Maria and Sam both have the same percentage of winning then she is sure to go broke. But that is not the case here because Maria has a better chance of winning each bet.

]]>Seems like a great game for Maria right?

But what if Sam had an infinite amount of money, and Maria has a finite amount. ($100 for example.) What if these players played this game an infinite amount of times. (let's assume they are immortal and they never lose the dice.)

Does Maria eventually go broke?

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