Suppose that { An }n20 is a Markov Chain (MC) with state space S = {0, 1, 2, 3} and transition matrix P: 1/2 1/2 0 0 0 B 0 P = 0 0 B 0 0 1 0 where 0 00. 4. For n > 0, define the probability distribution of Xn by 7(7) = [P(X,. = 0), P(X, = 1), P(X,. = 2), P(X,. = 3)] . Let a = 8 =1/2. Specify 70) so that, for all n > 0, X, has the same probability distribution, i.e., 7(") = (# 1)=... = x(0). 5. Suppose that a system is modelled by the MC {X, }zo with a = 8 = 1/2, and a gambler plays against a machine based on the state of the system: in each game, he loses $1 if the system is at states 0, 1, 2 or he will win $5 if the system is at state 3. Find the gambler's average winning in 100 such games