{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] = ?
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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...