Consider a Markov Chain $\displaystyle X_0, X_1,...,$ with state space {1,2,3}, initial distribution $\displaystyle \pi^{0} = (1/3,1/3,1/3) $ and one-step transition matrix P where P is the attached image.

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(1) $\displaystyle \mathbb{P} (X_2 | X_0 = 1)$

(2) $\displaystyle \mathbb{P}(X_2 = 3)$

(3) $\displaystyle \mathbb{E}(X_3)$

(4) $\displaystyle \mathbb{P}(X_0=1 | X_3 = 3)$