This question is really throwing me off:
Lets say there's two players, A and B. Each game consists of betting \$1. Gameplay ends when one player has all of the money. Player A starts with \$3, B starts with $5. If Player A has probability of winning 2/3, what's the probability that Player A will win the whole thing?
I understand that we want to take into account the probability that player A will win 5 in a row: $(2/3)^5$ and add it to the probability that after some games we'll be back at the start. Not sure where to go forward though.
For $k\in\left\{ 0,1,\dots,8\right\} $ let $p_{k}$ denote the probability that $A$ will win the whole thing if $A$ starts with $k$ dollars.
Then $p_{0}=0$, $p_{8}=1$ and $p_{k}=\frac{2}{3}p_{k+1}+\frac{1}{3}p_{k-1}$ for $1\leq k\leq7$.
Do you understand why?
Now find a solution for these equations. You are actually looking for $p_3$.