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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.

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  • $\begingroup$ 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 $P(X_0=X_1=1 \, | \, X_0=X_1)$. $\endgroup$ – Just_to_Answer Jan 9 at 4:06
  • $\begingroup$ I am not yet exposed to Matrix calculations. Is there another way $\endgroup$ – Math_Guy Jan 9 at 4:59
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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 $$\begin{eqnarray} &&P(\text{consecutive days are sunny}\;|\;\text{consecutive days have the same weather})&\\&&=\frac{P(\text{consecutive days are sunny})}{P(\text{consecutive days are sunny})+P(\text{consecutive days are cloudy})}\\&&=\frac{\frac{3}{7}}{\frac{3}{7}+\frac{2}{7}}=\frac{3}{5}. \end{eqnarray}$$

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