Explain the conditional probability part in depth

 

d. By the Markov property and time homogeneity, we have P{X3 = 1, X = 1|Xo = 1} = = = = = P{X3 = 1, X = 1, Xo = 1} P{Xo = 1} P{X3 = 1|X2 = 1, Xo = 1} x P{X3 = 1|X = 1} x P{X = 1|Xo = 1} P{X 1|Xo = 1} x P{X = 1|Xo = 1} = P{X = 1, Xo = 1} P{Xo = 1}