Find the Probability That Three Circle Points Fit in a Semicircle

What is the probability that three points on a circle all lie within some semicircle? Derive the answer and describe a Monte Carlo check that does not double-count successful trials. ### Constraints and Clarifications For the practice model, sample the three points independently and uniformly on the circumference. The source does not fully specify the sampling distribution; points sampled inside a disk or by another mechanism would be a different problem. Closed versus open semicircles differ only on probability-zero boundary events in this continuous model. ### What a Strong Answer Covers Use rotational symmetry or a disjoint-event argument, explain why the candidate semicircles do not double-count ordinary configurations, and specify a simulation success condition. ### Follow-up Questions - How can sorted angular gaps identify whether a semicircle contains all points? - How does the argument extend to n independent uniform circle points?