Bag I contains $\displaystyle 3$ red and $\displaystyle 4$ black balls, Bag II contains $\displaystyle 5$ red and $\displaystyle 2$ black balls. Two balls are transferred at random from Bag I to Bag II and then a ball is drawn at random from Bag II. Find the probability that the drawn ball is red in colour.

E1: Two Balls transferred from Bag I are Red. E2: Two Balls transferred from Bag I are Black. E3: Two Balls transferred from Bag I are Red and Black. A: Ball drawn from Bag II is Red $$\begin{gathered} \mathbf{P}\left(\mathrm{E}_{1}\right)=\frac{1}{7}, \mathbf{P}\left(\mathrm{E}_{2}\right)=\frac{2}{7}, \mathbf{P}\left(\mathrm{E}_{3}\right)=\frac{4}{7}, \mathbf{P}\left(\mathrm{~A} \mid \mathrm{E}_{1}\right)=\frac{7}{9}, \mathbf{P}\left(\mathrm{~A} \mid \mathrm{E}_{2}\right)=\frac{5}{9}, \mathbf{P}\left(\mathrm{~A} \mid \mathrm{E}_{3}\right)=\frac{6}{9} \\ \mathbf{P}(\mathrm{~A})=\frac{1}{7} \cdot \frac{7}{9}+\frac{2}{7} \cdot \frac{5}{9}+\frac{4}{7} \cdot \frac{6}{9}=\frac{41}{63} \end{gathered} $$