2024 Gauss 8 Problem 24

A network of pathways lead from a single opening to three bins, labelled A, B, C as shown. If a ball is dropped into the opening, it will follow a path and land in one of the bins. Every time a path splits, it is equally likely for the ball to follow either of the downward paths. Ellen drops two balls, one after the other, into the opening. What is the probability that the two balls land in different bins?

\textbf{(A)}\ \frac{17}{32}\quad \textbf{(B)}\ \frac{27}{50}\quad \textbf{(C)}\ \frac{25}{64}\quad \textbf{(D)}\ \frac{1}{3}\quad \textbf{(E)}\ \frac{15}{32}

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.