Cloudy and Sunny Days 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.

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?