Perpendicular bisectors in a circle

Geometry Level 5

Let $S$ be a set of $31$ equally spaced points on a circle centered at $O$ , and consider a uniformly random pair of distinct points $A$ and $B$ ( $A, B \in S$ ). The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed as $\frac{m}{n}$ , where $m,n$ are relatively prime positive integers. Find $m+n$ .

This problem is posed by Muhammad A.