Hi;

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.