Law of Total Probability in Markov Chains

Question

I'm reading about Markov Chains and have come across the following:

$ P_x (X_2 = y) = \sum\limits_{z\in \mathbb S} P_x (X_1 = z).P_x(X_2 = y|X_1 = z) $

where $ P_x (X_1 = z) = p(X_1 = z|X_0 = x) $

which is obtained through the law of total probability. I don't quite understand how this law has been applied to obtain the above Equation.

Eventually the following is obtained: $ \sum\limits_{z\in \mathbb S}p (x,z).p(z,y) $

which is apparently equal to $p^2(x,y)$. I thought it should be equal to $p^2(z,y)$, as we are going from x to y?

Thanks

Answer 1

Law of total probability use because of you can travel from $ x $ to any other states in state space. So we take addition over all states, $ P_x(X_2= y) = \sum_{z\in S} P_x(X_1= z).P_x(X_2= y|X_1= z)$

And secondly $ P^2(x,y)$ means in two steps you are going from $ x$ to $y$. And it is related to product of transient probability matrix.