\begin{align*} P(\text{Rain | Forecast Rain}) & = \frac{\frac{5}{365} \cdot 0.9}{\frac{5}{365} \cdot 0.9 + \frac{360}{365} \cdot 0.1}\\ & = \frac{5 \cdot 0.9}{5 \cdot 0.9 + 360 \cdot 0.1}\\ & = \frac{45}{45+360} \approx 0.111 \end{align*}