The expected number of sunny days in a year

Question

Suppose there are two types of weathers. Sunny and Rainy.
The probability that a sunny day is followed by a sunny day is 70% and followed by a rainy day is 30%.
The probability that a rainy day is followed by a rainy day is 60% and followed by a sunny day is 40%.
In a year (365 days), how many days do we expect to be sunny?

Based on the question above, I only get the transition matrix
$
\begin{bmatrix}
0.7 & 0.3\\
0.4 & 0.6
\end{bmatrix}
$
May I ask how do I calculate the expected number of sunny days in a year?
Thanks in advance.

Answer 1

Taking that the steady state probability exists, let these probabilitie be s for sunny and r for rainy, then one more iteration won't change these probabiliies, hence

$s*0.7 + r*0.4 = s$

$s*0.3 + r*0.6 = r$

$s+r=1$

which yields $s = \frac47, r = \frac37$

The required answer $=\frac47\times 365$