# Finding the probability of a Markov Chain.

If $$(X_n)_{n≥0}$$ is a Markov chain on $$S = {1, 2, 3}$$ with initial distribution $$α = (1/2, 1/2, 0)$$ and transition matrix

$$\begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix},$$

then $$P(X_2 = 2) = ?$$.

My solution:

$$X_1 = \begin{bmatrix} 1/2&1/2&0 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix}$$

$$X_2 = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} = \begin{bmatrix} 3/8&3/8&1/4 \end{bmatrix}$$

So, $$P(X_2=2) = 3/8$$

Is this solution correct?

Why or why not?

• Yes, this is correct. – Math1000 May 6 at 20:26
• @Math1000, why is it correct? – user366312 May 6 at 20:34