I am going to give a talk about Markov Chains to a group of high school students with very basic knowledge of probability. Because all of them have been introduced to EE, I wanted to demonstrate through an example the connection between random walks and electrical networks.
In particular, my example will be Gambler's ruin for $n=5$ and the corresponding electrical circuit (which is 5 equal resistors in series connected to a 1 volt battery). You can read about it in the first few pages of this paper.
I will follow a pretty similar process to the one outlined in the paper. I will derive a recurrence relation for $p(k)$, the probability that the gambler will reach $5$ dollars given that he started with $k$, and then I will derive the same recurrence (with same boundary conditions) for the voltage in the nodes between the resistors. Then I will conclude that the two functions must be the same and proceed to compute them.
My question regards the intuition behind this. I want to convey them some intuition about this but without getting into harmonic functions. I am thinking of connecting the transition probabilities of the Markov chain with the conductance of the circuit (i.e. saying something along the lines of "a higher transition probability makes us more likely to select this route, which is equivalent to a circuit element having higher conductance"). But I find this too hand-wavy, even for this context. Could someone suggest a more precise/detailed description of the analogy between these two concepts?