Step 1:

Given Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball drawn is red in color.

Let $E_1$ be the event that a red ball is transferred from Bag I to Bag II, and $E_2$ be the event that a black ball is transferred from Bag I to Bag II. Let A be the event that the ball drawn is red.

$P (E_1) = \large\frac{3}{3+4} = \frac{3}{7}$

$P(E_2) = \large\frac{4}{4+3} = \frac{4}{7}$

Step 2:

$P \large(\frac{A}{E_1}) = \frac{5}{10} = \frac{1}{2}$

$P \large(\frac{A}{E_2}) = \frac{4}{10} = \frac{2}{5}$

$P\left(\large \frac{E_2}{E}\right ) = \Large \frac{P\left(\frac{E}{E_2}\right ). P(E_2)} {\sum_{i=1}^{4} (P\left(\frac{E}{E_i}\right ).P(E_i))}$.

$\Rightarrow \large\;\frac{\frac{4}{7}\;\times\;\frac{2}{5}}{\large\frac{3}{7}\;\times\;\frac{1}{2}\;+\;\frac{4}{7}\;\times\;\frac{2}{5}}=\large\;\frac{16}{31}$