The Problem in the "Nerd Sniping" xkcd Comic

A few days ago I saw this comic reposted, and I thought: wait! Unlike every prior time I have seen this comic, I actually know how to solve this now! One thing which I often find really cool, and which I don’t think comes up a lot in most people’s mathematical education, is when you can learn something about some thing non-random by analysing a random process. So, let me show you a way to find out the resistance between two points in a circuit by instead finding out the number of times someone randomly walking around that circuit will step on those points.