Jane and John are playing a card game. Jane is an expert and, on each game, has probability p = 0.7 of beating John. Suppose that Jane and John each begin with 10 dollars and agree that, on each game, the loser pays the winner $1. They agree to play until one of them is bankrupt. What is the probability that it will be John who goes bankrupt?
My approach to this question is that this seems to be a Dicrete Time Markov Chain. In that case, I let $X_n =$ the amount of money John has at game $n$. In that case, I am trying to find $\pi_0$ My logic is that am I trying to find the long run probability that John is in state 0, i.e. he is bankrupt. Finding this value is very complicated as there are 20 possible states for John to be in. Is my approach correct? Or is there a simpler way to do this?