Expected time for an upward-biased walk to gain 1

A particle on the integers begins at $0$. Each step it moves $+1$ with probability $p=\tfrac23$ and $-1$ with probability $\tfrac13$. Find the expected number of steps until it first reaches $1$ (answer $-1$ if this is infinite).

$3$. Let $m$ be the expected time to rise one level. After a down step you must rise two levels, taking $2m$ on average, so $m=1+(1-p)\cdot 2m$, giving $m=\tfrac{1}{2p-1}=3$. The drift argument agrees: average speed $\tfrac13$ per step.