I am trying to work on this problem, pasted below:
Let's say we start at point 1. Each successive point you have a, say, 2/3 chance of increasing your position by 1 and a 1/3 chance of decreasing your position by 1. The walk ends when you reach 0.
The question, what is the probability that you will eventually reach 0?
The selected answer solves the following equation
$$ r = \frac13 + \frac23 r^2$$
The roots are $r = 1$ and $r = 1/2$.
I'm struggling to understand how the answer rules out $r = 1$. I'm not sure what the $\tau_{n+1}$ meant in the answer. Alternatively, can someone explain to me how to rule out the root of $r=1$, with or without following the answer?
