Three random points in a semicircle

Question

Three points are placed independently and uniformly at random on the circumference of a circle. What is the probability that all three lie within some semicircle? Generalize to n points.

Accepted Answer

The trap is that the semicircle isn't fixed in advance, it can be any half of the circle. The fix: anchor candidate semicircles at the points themselves. For each point i, let Ai be the event that the semicircle starting at point i and running clockwise contains all n points. Two observations: 1. If all points fit in some semicircle, exactly one Ai occurs, the event forces a "first" point (the one with nothing in the half-circle counterclockwise behind it), and with probability 1 no ties occur. So the Ai are disjoint and their union is the target event. 2. P(Ai) is easy: given point i's locat…

Three random points in a semicircle

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