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?

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    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 1/2n1. 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/2n1 (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.

    ronit Guru Answered on 5th August 2015.
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