A science club has two urns. Urn I holds 5 red balls and 3 black balls…

Step 1. The cases. The transferred ball is red with probability $\dfrac{5}{8}$ and black with probability $\dfrac{3}{8}$.

Step 2. Urn II after the transfer. After a red transfer, the chance of drawing red from Urn II is $\dfrac{5}{11}$. After a black transfer, it is $\dfrac{4}{11}$.

Step 3. Both red. $\dfrac{5}{8} \times \dfrac{5}{11} = \dfrac{25}{88} \approx 0.2841$.

Step 4. The second ball is red. Add the other path: black then red is $\dfrac{3}{8} \times \dfrac{4}{11} = \dfrac{12}{88}$. So $P(\text{red}) = \dfrac{25}{88} + \dfrac{12}{88} = \dfrac{37}{88} \approx 0.4205$.

Step 5. The second ball is black. $1 - \dfrac{37}{88} = \dfrac{51}{88} \approx 0.5795$.