What are the possible ways that the urns could end up in the same configuration? Say he chooses a white ball from urn 1 (this happens with probability $\frac{3}{5}$). He then places it into urn 2 and then removes a random ball from urn 2. In order for the configuration to stay the same, the ball he removes must be white. This happens with probability $\frac{3}{4}$ because there are now 3 white balls and one black ball in urn 2. Lastly, he places the white ball in urn three, which then has two of each color ball. The probability that he then removes a white ball is $\frac{1}{2}$. Since these events are all independent, the probability that they all happen in succession is $\frac{3}{5}\cdot\frac{3}{4}\cdot\frac{1}{2}=\frac{9}{40}$. To finish the problem, do the same analysis supposing he removes a black ball from urn 1 and combine the answers.