Problem Analysis #3

Return a single integer only. The problem has been revised so that no final answer is an irrational number, a parameterized expression, a range, a proof, a drawing, a graph, a locus equation, or a region description.

Let $n=9$. Choose four distinct vertices of a regular $n$-gon uniformly at random. Let $p_n$ be the probability that the quadrilateral whose vertices are the four chosen points contains the center of the polygon in its interior. If $p_9=p/q$ in lowest terms, find $p+q$.