Let X₀, X₁, X₂, ... be a Markov chain with state space S = {1,2,3}, transition probabilities
P = ( 0 1/2 1/2 )
( 1 0 0 )
( 1/3 1/3 1/3)

and initial distribution aᵀ = (1/2,0,1/2). Find the following probabilities:
(a) P [X₂ = 1 X₁ = 3],
(b) P [X₁ = 3, X₂ = 1],
(c) P [X₁ = 3 X₂ = 1],
(d) P [X₉ = 1|X₁ = 3, X₄ = 1, X₇ = 2].