Lizzy eats out every Saturday and each week she chooses between three restaurants: a Thai restaurant (T), an Indian restaurant (I) and a Chinese restaurant (C). Each Saturday, her choice of restaurant depends on which restaurant she chose the previous Saturday. The pattern of Lizzy's restaurant choices can be modelled by a Markov chain with the following transition matrix. T (1 2 0 P = I C (a) Last Saturday, Lizzy ate at the Chinese restaurant. What is the probability that she will eat at the Thai restaurant next Saturday and the Indian restaurant the following Saturday? (b) What is the probability that Lizzy will eat at the Chinese restaurant this coming Saturday, given that she ate at the Thai restaurant the Saturday before last? (c) Suppose that this Markov chain has a limiting distribution. Find this limiting distribution without using an iterative method. In the long run what proportion of Saturdays will Lizzy choose the Thai restaurant?

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