Biased walk with up-probability one third reaching 7

A particle starts at $0$ on the integer line. At every step it moves up by one with probability $p=\tfrac13$ and down by one with probability $\tfrac23$, independently of the past. What is the probability that the particle ever visits position $7$?

$\tfrac{1}{128}$. The chance of ever climbing one level from where you are is $\tfrac{p}{1-p}=\tfrac12$ (from the equation $q=p+(1-p)q^2$, taking the smaller root). Climbing seven levels needs seven independent such successes, so the answer is $\left(\tfrac12\right)^7$.