Let $S_n$ be a biased random walk with $P(S_{n+1} = S_n +1) = p >1/2$ and $P(S_{n+1} = S_n -1) = 1-p$. Suppose $S_0 = 1$ and let $\eta= \inf \{ n \colon S_n = 0\}$. Possibly $\eta = \infty$, so take $A = \{ \eta <\infty\}$. I have been told that the law of $S_n$ conditioned on $A$ is a biased random walk with the reverse bias (i.e. decreases with probability $p$ and increases with probability $1-p$ until reaching $0$).
I imagine this can be shown through some brute force computation, which I would be interested to see, but I am more interested if there is there a high-level way to prove this.