Evaluate conditional probability given 1 step transition matrix?

Question

Evaluate P(X2 = 0, X3 = 4, X5 = 4, X6 = 5 | X1 = 0). Justify your answer.

I am given a 6x6 one-step transition matrix for the actual numeric answer but I just need help in simplifying the conditional probability to work it out. I've used the rule P(A,B|C) = P(A|B,C) to split the probability and then use the ideas of a Markov Chain to reduce them to something like this:

P(X6=5|X5=4) * P(X5=4|X1=0,X3=4,X2=0) * P(X3=4|X2=0) * P(X2=0|X1=0)

All of the probabilities are very simple to calculate from the matrix, although I am confused about the bolded part. Can you please let me know if

a) I simplified the probabilities correctly

b) What to do with the bolded probability to simplify it?

Thanks

Answer 1

$P(X_5=4|X_1=0,X_3=4,X_2=0)=P(X_5=4|X_3=4)$ which is a two-step transition probability. You have to compute the square of the transition matrix or write $P(X_5=4|X_3=4)=\sum_i P(X_5=4||X_4=i)P(X_4=i||X_3=4)$.