In Imaginationland, each person has exactly one child. Suppose that a child of a pro video gamer is also a pro video gamer with prob- ability 0.7, or an amateur video gamer with probability 0.3. Suppose that the child of an amateur video gamer is a pro video gamer with probability 0.1, or an amateur video gamer with probability 0.5, or a non-gamer with probability 0.4. Suppose that the child of a non- gamer is a pro video gamer with probability 0.1, or an amateur video gamer with probability 0.5, or a non-gamer with probability 0.4. (a) By drawing a suitable directed graph with probability edge-weights, model the described scenario as a three-state discrete-time Markov chain. (4 marks) (b) Write down the transition matrix P corresponding to the Markov chain that you drew in (a). (4 marks) (c) Based on your answer to (b), calculate the probability that the great grand-child of a non-gamer is a pro video gamer. (6 marks) (d) List the communication classes of the Markov chain that you drew in (a). For each communication class, determine with jus- tification whether it is recurrent or transient. Please state in full any relevant general results or definitions. (6 marks) (e) Compute, with justification, a stationary distribution for the Markov chain that you drew in (a). Is the stationary distribution unique? In the long run, what proportion of people in Imaginationland are pro video gamers? (5 marks)