I have a random walk at position i that goes to i+2 with probability 1/2 and goes to i-1 with probability 1/2. We start at position 0 and are absorbed by position -1. What is the probability of being absorbed in an infinite random walk?
I am trying to solve this by thinking about recurrence relationships where
$$f(x) = \frac{1}{2} f(x-1) + \frac{1}{2} f(x+2)$$ and $$f(-1) = 1$$
But I haven't gotten anywhere. Can anyone provide insight?
Are you familiar with the standard method of solving linear recurrences with constant coefficients? You can write your equation as $$f(n+2)-2f(n)+f(n-1)=0\ ,$$ and the usual method applies. In this case you will have a cubic characteristic equation, but one of the roots will be clear if you look closely.
You will need conditions in order to determine the unknown constants in your solution. The value of $f(-1)$ should be obvious, also consider what happens when $n\to\infty$.
If you have not studied this topic just ask Google, there will be many explanations online.