2019 AMC 12B Problem 13

A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin k is 2^{-k} for k = 1,2,3.... What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.