Suppose that X 0 , X 1 , X 2 , ... form a Markov chain on the state space {1, 2}. Assume that P(X 0 = 1) = P(X 0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the...

Suppose that X₀, X₁, X₂, ... form a Markov chain on the state space {1, 2}. Assume that P(X₀
= 1) = P(X₀
= 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q₁₁
= 1/2, Q₁₂
= 1/2, Q₂₁
= 1/3, Q₂₂
= 2/3.
(a) Find P(X2 = 1).
(b) Find the conditional probability P(X2 = 1|X1 = 1).
(c) Find the conditional probability P(X1 = 1|X2 = 1).
(d) Find lim_n→∞
P(Xn = 1).