The question:
Find a probability that a year chosen at random has 53 Mondays.
Now I know in a non-leap year, probability of getting 53 Mondays is $\frac{1}{7}$ and in a leap year, probability of getting 53 Mondays is $\frac{2}{7}$. Now knowing that leap year occurs after every 4 years, I felt the desired probability is $$\frac{1}{7}\times\frac{4}{5}+\frac{2}{7}\times\frac{1}{5}=\frac{4}{35}+\frac{2}{35}=\frac{6}{35}$$ However the solution given without any explanation is $\frac{5}{28}$. How is that? (I might be applying stupid logic with that $\frac{4}{5}$ and $\frac{1}{5}$).
A leap year occurs every fourth year. That means your weights must be $\frac34$ and $\frac14$ resp. yielding $$\frac17 \cdot \frac34 + \frac27 \cdot \frac14 = \frac5{28}$$ The pattern of leap years is $$\Box\Box\Box\times\Box\Box\Box\times\Box\Box\Box\times\cdots$$