Consider the Markov Chain with state space $S=\{v,w,x,y,z\}$, transition matrix below:
$$\left[\begin{array}{cccccccccc} 0 & 0.4 & 0.6 & 0 & 0\\ 0 & 0.5 & 0.5 & 0 & 0\\ 0 & 0 & 0 & 0.1 & 0.9\\ 0 & 0 & 0& 0.2 & 0.8\\ 0.7 & 0 & 0.3 & 0 & 0 \end{array}\right]$$ (where the matrix rows/columns correspond to the states in alphabetical order), and $X_0=v$.
1) What is $P(X_1=x, X_2=z, X_3=v)$?
For this question, do I multiply all the states starting from state x -> state z -> state v?
which I get: $$0.9 \times 0.7 = 0.63$$ Am I on the right track?
2) Find the probability distribution of $X_4$.
Any hints for this question please?
3) Find $P(\{X_2=w\} \cup \{X_3=x\})$ and $P(X_2=x|X_4=z)$
Im completely getting lost. I'm assuming that the first one is finding the union of $X_2$ and $X_3$ and the second question is finding the probability of $X_2$ given $X_4$, I am not sure what these "X" values are. Anyone could you please help me or give me a hint on these questions?