{Xn, n > 0 } is a Markov Chain, such that Xn is an element of {1, 2},

P[Xn+1 = 2 | Xn = 1] = a, which is a probability between (0,1)

P[Xn+1 = 1 | Xn = 2] = b, which is a probability between (0,1)

P[X5 = 1 | X1 = 1] = ?

P[Xn+1 = 2 | Xn = 1] = a, which is a probability between (0,1)

P[Xn+1 = 1 | Xn = 2] = b, which is a probability between (0,1)

P[X5 = 1 | X1 = 1] = ?

Please find...

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Given that,

P(Xn+1=2|Xn=1)=a ; i.e. P(Xn+1=1|Xn=1)=1-a

P(Xn+1=1|Xn=2)=b ; i.e. P(Xn+1=2|Xn=2)=1-b

The transition matrix is

P=

where pxy=P(X1=y|X0=x)

The required probability is the (1,1) th element...