2021 Gauss 8 Problem 8

Initially, there are 5 red beans and 9 black beans in a bag. Then, 3 red beans and 3 black beans are added to the bag. If one bean is randomly chosen from the bag, what is the probability that this bean is red?

\textbf{(A)}\ \frac{3}{8}\quad \textbf{(B)}\ \frac{2}{5}\quad \textbf{(C)}\ \frac{4}{5}\quad \textbf{(D)}\ \frac{5}{8}\quad \textbf{(E)}\ \frac{8}{17}

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 Gauss contests here.