Random walking and the expected value I was asked this question at an interview, and I didn't know how to solve it. Was curious if anyone could help me.
Lets say we have a square, with vertex's 1234. I can randomly walk to each neighbouring vertex with equal probability. My goal is to start at '1', and get back to '1'. How many average walks will I take before I return back to 1?

By symmetry, the unique invariant probability measure $\pi$ for this Markov chain is uniform on the four states. The expected return time is therefore $\mathbb{E}_1(T_1)=1/\pi(1)=4.$ 
This principle is easy to remember and can be used to solve other interesting problems.
