# Freedonia Rainy and Sunshine

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 $$\frac 23$$. Given that a consecutive Saturday and Sunday had the same weather in Freedonia, what is the probability that that weather was sunny?

So far, I have tried to do (Sunny)/(Sunny + Cloudy) which ended up giving me $$9/17$$. This though is wrong.

I know this question has been asked before but none of them gave me a complete answer.

• It might be easier to think about this as a Markov process $\{X_n\}$ with states $0=$cloudy and $1=$sunny. You can identify the transition matrix from the given probabilities. Let $n=0$ and $n=1$ represent Saturday and Sunday, respectively. Then you want $P(X_0=X_1=1 \, | \, X_0=X_1)$. – Just_to_Answer Jan 9 at 4:06
• I am not yet exposed to Matrix calculations. Is there another way – Math_Guy Jan 9 at 4:59

Let $$p=$$ the probability that Freedonia is sunny on a day and $$q=$$ the probability that Freedonia is cloudy on a day. Then the given condition implies the following equilibrium condition holds: $$\begin{eqnarray} p&=&\frac{3}{4}p+\frac{1}{3}q,\\ q&=&\frac{1}{4}p+\frac{2}{3}q. \end{eqnarray}$$ Using $$p+q=1$$, we can solve $$p=\frac{4}{7}$$ and $$q=\frac{3}{7}$$. Now, the probability that consecutive days are sunny days is $$p\cdot \frac{3}{4}=\frac{3}{7}$$, and probability that consecutive days are cloudy is $$q\cdot\frac{2}{3}=\frac{2}{7}$$. By the definition of conditional probability, this gives $$\begin{eqnarray} &&P(\text{consecutive days are sunny}\;|\;\text{consecutive days have the same weather})&\\&&=\frac{P(\text{consecutive days are sunny})}{P(\text{consecutive days are sunny})+P(\text{consecutive days are cloudy})}\\&&=\frac{\frac{3}{7}}{\frac{3}{7}+\frac{2}{7}}=\frac{3}{5}. \end{eqnarray}$$