2016 Fermat Problem 21

Box 1 contains one gold marble and one black marble. Box 2 contains one gold marble and two black marbles. Box 3 contains one gold marble and three black marbles. Whenever a marble is chosen randomly from one of the boxes, each marble in that box is equally likely to be chosen. A marble is randomly chosen from Box 1 and placed in Box 2. Then a marble is randomly chosen from Box 2 and placed in Box 3. Finally, a marble is randomly chosen from Box 3. What is the probability that the marble chosen from Box 3 is gold?

\textbf{(A)}\ \frac{11}{40}\quad \textbf{(B)}\ \frac{3}{10}\quad \textbf{(C)}\ \frac{13}{40}\quad \textbf{(D)}\ \frac{7}{20}\quad \textbf{(E)}\ \frac{3}{8}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.