Suppose we start at State $1$, directly to the right of a State $0$. What's the probability that we ever get to State $0$, if we take n steps to the right with probability p, 1 step to the left with probability q, and stay forever at State i (the current state) with probability r.
I know how to construct the solutions to this question but I don't know the why. I would appreciate a basic, intuitive, and logical way to get these equations.