Points on a circle
Given N points randomly distributed around the circumference of a circle, what is the probability that all N points lie on the same semi-circle?
Suppose that point i has angle 0 (angle is arbitrary in this problem) — essentially this is the event that point i is the “first” or “leading” point in the semicircle. Then we want the event that all of the points are in the same semicircle — i.e., that the remaining points end up all in the upper halfplane.
That’s a coin-flip for each remaining point, so you end up with 1/2n−1. There’s n points, and the event that any point i is the “leading” point is disjoint from the event that any other point j is, so the final probability is n/2n−1 (i.e. we can just add them up).
A simple check for this answer is to notice that if you have either one or two points, then the probability must be 1, which is true in both cases.
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