Past exam of the options I course of the University of Oxford / 2011 / M / 2 / d / Solution

With probability $2/3$, the chain starts at $+1$ and $T=1$. Conditional on starting at $-1$, it takes a geometric number $G$ of transitions to first reach $+1$, with success probability $1-p$, so $T=1+G$. The probability generating function is thereforeThe parameter $q$ is absent because the process stops at its first visit to $+1$. At $s=1$ the expression equals one, confirming eventual hitting with probability one.