I'm trying to understand Markov Chains and have across the following in a book:
$ \sum\limits_{y=0,1,....m−1}p(x,y)P(T_A<T_B|X_0=x,X_1=y) $ which then becomes the following, under the Markov Property
$ \sum\limits_{y=0,1,....m−1}p(x,y)P(T_A<T_B|X_0=y) $
I understand that the Memoryless property of Markov Chains is being referred to, but I don't understand how it is being used to get X_0=y
I basically don't understand how to relate the above to the property,
$ P(X_{n+1}=x_{n+1}|X_0 = x_0, X_1 = x_1,......X_n = x_n) = P(X_{n+1}=x_{n+1}|X_n = x_n) $
Thank You