Prince Charming is coming to rescue Sleeping Beauty (SB) from her slumbering imprisonment with a kiss. However, SB sleeps in one of three positions: State 0: flat on her back, in which case she looks truly radiant; State 1: foetal position, in which case she looks less than radiant; State 2: foetal position and sucking her thumb, in which case she does not look radiant at all. Changes of position occur according to a Markov Chain with transition matrix
P = $\begin{bmatrix} 0 & 0.75 & 0.25 \\ 0.25 & 0 & 0.75 \\ 0.25 & 0.75 & 0 \end{bmatrix}$
$v_0 = 1/2$, $v_1 = 1/3$ $v_2 = 1$ What is the probability that Prince Charming finds SB looking truly radiant at the time he arrives? Naturally, SB has been sleeping for a very long time.
I started this question by finding the matrix G instead as it's continuous time and found
G = $\begin{bmatrix} -\frac{1}{2} & \frac{3}{8} & \frac{1}{8} \\ \frac{1}{12} & -\frac{1}{3} & \frac{1}{4} \\ \frac{1}{4} & \frac{3}{4} & -1 \end{bmatrix}$
However, I'm not sure how to finish the question from here, am I looking for $P_{00}(t) + P_{10}(t) + P_{20}(t)$ to find the total probabilities? If so how do I actually find all those? Just looking for some guidance to understand the next step or if I'm even on the right track.
I actually figured it out. I was on the right track the $G$ matrix, however the $P(t)$'s did not matter and should have focused on $\pi_0$ which is solved using the system of equations from $\pi G=0$ where we get
$-\frac{1}{2}\pi_0 + \frac{1}{12}\pi_1 + \frac{1}{4}\pi_2 = 0$
$\frac{3}{8}\pi_0 + -\frac{1}{3}\pi_1 + \frac{3}{4}\pi_2 = 0$
$\frac{1}{8}\pi_0 + \frac{1}{4}\pi_1 + -\pi_2 = 0$
$\pi_0 + \pi_1 + \pi_2 = 1$
solving and subsituting we get $\pi_0$ = 0.1944