A standard gambler's ruin markov chain set up is the following:
To find the probability $P_i$ that someone with \$i (i.e. at state i) will go up to state \$N before going to state $0$ (i.e. losing), we solve a recurrence $P_i = pP_{i-1} + qP_{i+1}$ and derive the solution using the standard methods in a textbook.
In these steps, we have the following equation:
$$P_i=P_1 \left(1+\frac{q}{p}+\left(\frac{q}{p}\right)^2+\cdots+\left(\frac{q}{p}\right)^{i-1}\right)$$
I was wondering, is there some sort of intuition for this equality? And perhaps more generally, is there some sort of intuition for what $\frac{q}{p}$ means in this context?
When I say intuition, I am thinking of something maybe similar to the answer in this post: How many flips of a fair coin does it take until you get N heads in a row?
(Granted, the context is quite irrelevant to this. But you can interpret each term individually. Can you do the same here?)
