Consider the transition matrix below of a homogeneous Markov chain:
(a) Given that the process starts in state 1 at time 0 what is the probability that at time 2 it will be in state 3?
(b) In the long run what is the probability that the process will be in state 1, in state 2, in state 3?
(c) Describe some real-world situation for which the Markov chain above might be a crude model. Note particularly the last row of the transition matrix. Comment briefly on why and under what circumstances your Markov-chain model might fail to be a good or even approximate representation of the physical problem you consider.