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Suppose we do a sequence of Bernouill(p) trials,
we call them coin tosses, with a p-coin.
A wins the toss if the coin comes up heads,
and B wins if tails.
If A wins he takes 1 from B,
if not the inverse happens.
The overall game is played until
one of the players goes bankrupt.
Then we do a ``one-step'' analysis,
we can show that:
Now we use the two boundary conditions:
and the recurrence above leads to the following
conclusions (using the sum of the geometric series of term