Gambler's Ruin - Random Walks
Hi all, I was just doing a few exercise questions relating to Gambler's Ruin more specifically random walks pobabilites. I've come across this one particular problem im not too sure off. I cannot seem to make sense of the reflecting boundaries mentioned in the question. I was hoping someone could explain to me what's to be done in this situation. This is not a graded and just a exericse relating to gamblers ruin.
Consider a "random walker" on positive integers which moves
according to the following rules: starting from 1, it takes a step -1 with
probability p = 2/3 and a step +2 with probability q = 1 - p = 1/3. The
boundary at 0 is reflecting: if the walker ever reaches 0, it immediately (as
part of the same step) jumps back to 1. A player bets on the outcome as
follows: the bet is lost if the walker reaches point N + 1 without ever being
at point N, and the bet is won if the point N is reached before point N +1.
Let Pn be the probability that the bet is won if the starting point is n. Find
the dierence equation for Pn, and hence the probability of winning the bet
starting from point 1.