I have a system where a random walker is moving on $\mathbb{Z}$. However, at each point in $\mathbb{Z}$, there is a probability $q$ that an escape route exists along which the walker can escape. I want to find the expected time that the walker stays moving along the line before it escapes. At first I thought this would follow a $\operatorname{Geom}(p)$ distribution, where p is: \begin{align*} p &= \mathbb{P}(\text{leaving } \mathbb{Z})\\ &= \mathbb{P}(\text{an escape exists})\mathbb{P}( \text{taking escape} | \text{an escape exists})\\ &= \frac{1}{3}q. \end{align*}
With the factor of $1/3$ coming from the fact that with an escape route, a walker at $k$ could escape, move to $k+1$, or move to $k-1$, each with equal probability. However when I compared this to computer simulations it doesn't work very well. I think the issue is that the 'escape routes' are only determined once at the beginning (i.e. 5 has an escape route with probability $q$ - determined initially before the walker starts walking - but the above treats it as though the existence of a shortcut is determined anew each time you visit 5).
Could someone help me figure out what this distribution should be or if I can overcome this problem in the work I've done so far?