Off the top of my head, you would do it by getting the hitting probability of a snake from zero, and then the hitting probability of a snake from itself, and then take the expectation a la a geometric series.
And in case you don't know how to calculate the hitting probability, there's some notes on it here: http://www.statslab.cam.ac.uk/~rrw1/markov/M.pdf. More generally, Dexter's notes (should come up if you Google it) are great for a number of topics in maths.
For an example of how you'd go about calculating it for your specific problem, consider the following: squares 1, 2, 3 and 4, a snake from 3 to 1, a dice that rolls 1 or 2, and finishing at 4. Then, letting hi be the probability of hitting 3 from square i,
h3 = 1
h4 = 0
h2 = 1/2h3 = 1/2
h1 = 1/2h3 1/2h2 = 3/4
More generally, for any Markov chain, for set A to be hit, the hitting probabilities are the minimal solution to
hi = 1, I in A
hi = Σj pij hj, for transition probabilities pij
Hopefully that makes sense, and feel free to DM about any stats questions, as it was the focus of my undergrad :)
Thank you.
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