In Freedonia, every day is either cloudy or sunny (not both). If it's sunny on any given day, then the probability that the next day will be sunny is $\frac 34$. If it's cloudy on any given day, then the probability that the next day will be cloudy is $\frac 23$.
In the long run, what fraction of days are sunny?
How am I to solve this? My initial thought of 9/17 (obtained by $\frac{\frac{3}{4}}{\frac{3}{4}+\frac{2}{3}}$) was incorrect.
On
Use the law of total probability:
$$P(S)=P(S_2\mid S_1)P(S)+ (1-P(C_1 \mid C_2))(1-P(S))$$ where $P(S)$ is the probability of a sunny day, $P(S_2|S_1)$ is the probability of a sunny day following a sunny day and $P(C_2|C_1)$ is the probability of a cloudy day following a cloudy day.
\begin{gather} P(S)=3P(S)/4+ (1-2/3)(1-P(S))\\ P(S)/4=1/3 -P(S)/3\\ 7P(S)/12=1/3\\ P(S)=4/7\\ \end{gather}
This is a two-state transient Markov chain. The transition matrix is $$P = \left[\matrix{3/4 & 1/4\cr 1/3 & 2/3}\right] $$ Can you find a fixed probability vector for this matrix?