Question from STEP III Mathematics

[JaKiri]

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 p_a, p_b, p_c and p_d 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 p_a = 1/3 + (1/3)p_b + (1/3)p_d.

(ii) Determine p_a.

(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.