Two Conditional Probability In a city, 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 $\frac 23$. a) In the long run, what fraction of days are sunny? b) Given that a consecutive Saturday and Sunday had the same weather in the city, what is the probability that the weather was sunny? * * * For part a, I was thinking that the answer would be $(\frac{3}{4}+ \frac{1}{3})/2$ to get $\frac{13}{24}$ because the probability if it is sunny followed by another sunny day is $\frac34$ and the probability that it is cloudy followed by a sunny day is $\frac13$. For part b, I'm thinking of using casework, but I'm not sure.

Let's denote the given probabilities as $p_{s\mid s}=3/4$ and $p_{c\mid c}=2/3$. Then also $p_{s\mid c}=1/3$, which is the probability that a day is sunny if it follows a cloudy day.

For part (a)

Let's denote the equilibrium probability of that a day is sunny as $p_s$. At equilibrium this is a constant from day to day; so we want to solve $p_s= p_{s\mid s}\,p_s +p_{s\mid c}\,(1-p_s)$ for $p_s$.

$$p_s = \tfrac 3 4 p_s +\tfrac 1 3(1-p_s)$$

* * *

For part (b) use this equilibrium constant as the probability that Saturday was sunny, and work out $p_w$: the probability that the weekend was sunny given that both days had the same weather.

$$p_{w} = \frac{p_{s\mid s} p_s}{p_{s\mid s}p_s+p_{c\mid c}(1-p_s)} $$