Determining probabilities Markov Chain

Question

If I have a Markov Chain $X_0, X_1, X_2 \dots$ that has a transition probability matrix

$ \textbf{P} = \matrix{~ & 0 & 1 & 2 \cr 0 & 0.3 & 0.2 & 0.5 \cr 1 & 0.3 & 0.1 & 0.4 \cr 2 & 0.5 & 0.2 & 0.3\cr } $

and initial distribution $p_0=0.5$ and $p_1=0.5$. I am supposed to determine probabilites Pr{X2=0} and Pr{X3=0}.

Would I use conditional probability to solve this or is there a simpler way. I feel like I am overcomplicating thing because I am trying to use conditional probability but I don't have enough information to complete it.

Answer 1

$$Pr( X_2=0)=\sum_{i=0}^{2}Pr(X_2=0,X_0=i)=\sum_{i=0}^{2}Pr(X_2=0|X_0=i)Pr(X_0=i)$$ $$Pr(X_2=0|X_0=i)=(P^2)_{i,0}$$ These should be enough information for you to complete the question.