How is this problem related to electrical networks?

I am currently reading a book about stopping times, and just finished the chapter about electrical networks.

In the exercises section, there was this problem:

An Oregon professor has n umbrellas, of which initially k ∈ (0, n) are at his office and n − k are at his home. Every day, the professor walks to the office in the morning and returns home in the evening. In each trip, he takes an umbrella with him only if it is raining. Assume that in every trip between home and office or back, the chance of rain is p ∈ (0, 1), independently of other trips.

(a) Asymptotically, in what fraction of his trips does the professor get wet?

(b) Determine the expected number of trips until all n umbrellas are at the same location.

(c) Determine the expected number of trips until the professor gets wet.

I managed to solve this problem using elementary Markov chain tools. What strikes me as odd is that the exercise appeared in the chapter about electrical networks, and I couldn't find any way to apply them to solve the problem. In fact, there isn't even anything in this chapter about expected hitting times. I do assume there is some trick the authors intended that I find, given the placement of the problem and the fact that the elementary solution is suspiciously taxing. Can anyone here maybe see it?