2016 Cayley Problem 23

Suppose that PQRSTUVW is a regular octagon. (A *regular octagon* is an octagon with eight equal side lengths and eight equal interior angles.) There are 70 ways in which four of its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended infinitely in both directions, what is the probability that they will meet to form a quadrilateral that contains the octagon?

\textbf{(A)}\ \frac{1}{2}\quad \textbf{(B)}\ \frac{19}{35}\quad \textbf{(C)}\ \frac{37}{70}\quad \textbf{(D)}\ \frac{17}{35}\quad \textbf{(E)}\ \frac{18}{35}

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 Cayley contests here.