Q1. A Markov chain consists of a simple random walk taking place on a circle. The states consist of equally spaced points labelled 0, 1, 2, · · · , n in a clockwise direction. At each step of the random walk transition takes place as follows: (i) a clockwise step with probability p, (ii) an anticlockwise step with probability q (iii) p + q = 1. Find the probability that start at point 0, the mouse run a counter-clock circle. i.e. P(counter-clockwise circle|initial point=0).

Q2. A Markov chain consists of a simple random walk taking place on a circle. The states consist of equally spaced points labelled 0, 1, 2, · · · , n in a clockwise direction. At each step of the random walk transition takes place as follows: (i) a clockwise step with probability p_i, (ii) an anticlockwise step with probability q_i (iii) p_i + q_i = 1. Find the probability that start at point 0, the mouse run a counter-clock circle. i.e. P(counter-clockwise circle|initial point=0).