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3. Let {Xt: t=0,1,...} be a Markov chain with state space S = {1, 2, 3}, initial state distribution 77 = [0.5 0.5 0], and the following transition diagram: 0.3 0.5 1 0.7 0.5 0.4 3 0.2 0.4 Calculate the following probabilities, and show your working: = (a) P(X1 = 1) (b) P(X2 = 3|Xo = 3) (c) P(X2 = 2) (d) Unconditional probability of the trajectory 1, 2, 2 (e) P(X4 = 2|X3 = 3, X2 = 1, X1 = 2, X0 = 2)