Clue for exercise: What percent of the days are cloudy? > The weather on a given planet can be either cloudy or clear, with a constant probability. In $60\%$ of the cloudy days, the next day was clear and in $30\%$ of the clear days, the next day was cloudy. What percent of the days are cloudy? I defined the following: $\Omega = \\{((i,weather),(i+1,weather))\\}, 1 \leq i \leq 364$, weather is cloudy or clear. $ Cloudy_1 - \\{((i,cloudy),(i+1,weather))\\} $ $ Cloudy_2 - \\{((i,weather),(i+1,cloudy))\\} $ $ Clear_1 - \\{((i,clear),(i+1,weather))\\} $ $ Clear_2 - \\{((i,weather),(i+1,clear))\\} $ **I need to calculate:** $P(Cloudy_1 \cup Cloudy_2) $ **My conclusions from the questions are:** 1. $P(Clear_2 | Cloudy_1)=\frac{6}{10}$, $P(Cloudy_2 | Cloudy_1)=\frac{4}{10}$ 2. $P(Cloudy_2 | Clear_1)=\frac{3}{10}$, $P(Clear_2 | Clear_1)=\frac{7}{10}$ I played with the equations but looks like a dead end. **Any clue?** * * * # Don't give me the answer

**Hint:** if $A_i$, $B_i$ are the events that day $i$ is clear and cloudy, respectively, then $$P(A_{i+1})=P(A_{i+1}|A_i)P(A_i)+P(A_{i+1}|B_i)P(B_i)$$ using the law of total probability. Now if $a$ is the probability that a given day is cloudy, we have $P(A_{i+1})=P(A_i)=a$ and $P(B_i)=1-a$. Solve for $a$ using the equations you have already written down.