In Imagination land, each person has exactly one child. Suppose that a child of a pro video gamer is also a pro video gamer with prob- ability 0.5, or an amateur video gamer with probability 0.5. Suppose that the child of an amateur video gamer is a pro video gamer with probability 0.2, or an amateur video gamer with probability 0.2, or a non-gamer with probability 0.6. Suppose that the child of a non- gamer is a pro video gamer with probability p, or an amateur video gamer with probability q, or a non-gamer with probability 0.1. 

(a) Invent any reasonable values for p and q to apply for the entire question. Write down the transition matrix P describes the scenario.

(b) Calculate the probability that the grand-child of an amateur video gamer is a pro video gamer.

(c) List the communication classes of the Markov chain described in the question. For each communication class, determine with justification whether it is recurrent or transient. Please state in full any relevant general results or definitions.

(d) Compute, with justification, a stationary distribution for the Markov chain described in the question.

Is the stationary distribution unique? 

Is the Markov chain time-reversible? 

In the long run, what proportion of people in Imagine- nationland are non-gamers ?

(e) Compute the expected transition time (in number of generations) from non-gamer to pro video gamer.