Next step of a walk conditioned to hit 7

A particle walks on the integers, moving up by one with probability $p=\tfrac13$ and down by one with probability $\tfrac23$ at each step. It is now at $4$, and you are told that it will at some point reach $7$. Given this, what is the probability that its very next step takes it to $5$?

$2/3$. From $i$, the walk ever climbs $k$ levels with probability $(p/q)^k=2^{-k}$. By Bayes, $P(\text{up}\mid\text{hit }7)=p\cdot\tfrac{(1/2)^2}{(1/2)^3}=\tfrac13\cdot2=\tfrac23$. Conditioning on hitting the target flips the step probabilities to $q$ up and $p$ down.