Down-biased walk ever reaching plus one

A particle starts at $0$ on the integers and at each step moves up one with probability $p=\tfrac13$ or down one with probability $\tfrac23$, independently. What is the probability that it ever visits position $1$?

$\tfrac12$. Let $h$ be the probability of ever climbing one level. Either the first step goes up, or it goes down and the particle must climb two levels: $h=p+(1-p)h^2$. The roots are $1$ and $\tfrac{p}{1-p}$; with downward drift the smaller root applies, giving $\tfrac12$.