Players A and B each have $10 at the beginning of a game in which each player bets at each play, and the game continues until one player is broke. Suppose that Player A wins on a single bet with probability p = 0.45
What is the average duration of the game?
What is the probability that Player B wins the game?
So far I have created the transition matrix on the state space $S = \begin{Bmatrix}{0,1,2,3,4}\end{Bmatrix}$ assuming that it doesn't matter how much is being bet per round (where each state of the matrix represents have a multiple of 5 in their possession, i.e. $S_0 = 0, S_4 = 20$)
$\begin{bmatrix} 1 & 0 & 0 & 0 & 0\\0.55&0&0.45&0&0\\0&0.55&0&0.45&0\\0&0&0.55&0&0.45\\0&0&0&0&1 \end{bmatrix}$
I do not know how to proceed from here to answer the questions that are being asked. I would appreciate any help in this matter.
Edit: Each player bets \$1 each round so the state space would be $ S = \begin{Bmatrix}0,1,2,...,20 \end{Bmatrix}$ where each state represents a dollar in possession