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∈S). The probability that the perpendicular bisectors of OA and OB intersect strictly inside the circle can be expressed as m/n, where m,n are relatively prime positive integers. Find m+n.
To find the probability that the perpendicular bisectors of OA and OB intersect inside the circle, we need to examine the possible cases.
Case 1: OA and OB are diameters of the circle
In this case, the perpendicular bisectors of OA and OB will intersect at the center, which is inside the circle.
Case 2: OA and OB are not diameters of the circle
In this case, the perpendicular bisectors of OA and OB will intersect outside the circle.
Since there are 31 equally spaced points on the circle, there are (31 choose 2) = 465 possible pairs of distinct points A and B.
Out of these 465 possible pairs, the probability of Case 1 occurring is 1/31, as there is only one diameter passing through each of the points.
So the probability of Case 2 occurring is (465 - 1)/465 = 464/465.
Hence, the probability that the perpendicular bisectors of OA and OB intersect inside the circle is 1/31.
To find m+n, we have m = 1 and n = 31. Therefore, m+n = 1+31 = 32.