Four random points, one semicircle

Question

Four points are placed independently and uniformly at random on the circumference of a circle. What is the probability that all four lie within some common semicircle?

Accepted Answer

Fix the four points. For each point i, let Ai be the event that all four points lie in the half-circle starting at point i and sweeping clockwise. These events are mutually exclusive: at most one point can be the clockwise-most one of a group that all fit in a semicircle (the one from which the others span less than half the circle). For a fixed point i, each of the other 3 points independently lands in that specific half-circle with probability 	frac12, so P(Ai) = \left(	frac12ight)^3. There are 4 points, and the events cannot overlap, so P(	ext{all in a semicircle}) = 4 \cdot \left(	f…

Four random points, one semicircle

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