This is the question I liked the most from my STEP exam today.
13.
In a rabbit warren, underground chambers A, B, C and D are at the vertices of a square, and burrows join A to B, B to C, C to D and D to A. Each of the chambers also has a tunnel to the surface. A rabbit finding itself in any chamber runs along one of the two burrows to a neighbouring chamber, or leaves the burrow through the tunnel to the surface. Each of these three possibilities is equally likely.
Let pa, pb, pc and pd be the probabilities of a rabbit leaving the burrow through the tunnel from chamber A, given it is currently in chamber A, B, C or D, respectively.
(i) Explain why pa = 1/3 + (1/3)pb + (1/3)pd.
(ii) Determine pa.
(iii) Find the probability that a rabbit which starts in chamber A does not visit chamber C, given that it eventually leaves the burrow through the tunnel in chamber A.
GO.