2017 Fermat Problem 24

Eight teams compete in a tournament. Each pair of teams plays exactly one game against each other. There are no ties. If the two possible outcomes of each game are equally likely, what is the probability that every team loses at least one game and wins at least one game?

\textbf{(A)}\ \frac{1799}{2048}\quad \textbf{(B)}\ \frac{1831}{2048}\quad \textbf{(C)}\ \frac{1793}{2048}\quad \textbf{(D)}\ \frac{903}{1024}\quad \textbf{(E)}\ \frac{889}{1024}

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.