AMC 12 2020b Problem 16

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?

$\textbf{(A) } \frac16 \textbf{(B) }\frac15 \textbf{(C) } \frac14 \textbf{(D) } \frac13 \textbf{(E) } \frac12$