Three Points on a Circle

What this geometric probability question tests

This is an easy probability question commonly used to assess how a candidate approaches problems involving symmetry and geometric reasoning. Rather than relying on heavy computation, the question rewards clear thinking about what it means for points to satisfy a spatial constraint.

The key is to recognize that you can fix one point's position (by rotational symmetry) and then reason about the positions of the remaining points relative to it. The question probes whether you can translate a geometric constraint—"all three points fit within some semicircle"—into a probabilistic calculation over the positions of the other points. This kind of symmetry argument appears frequently in probability interviews and is faster and more elegant than brute-force integration.

• Symmetry and rotational invariance in probability
• Geometric constraints as order statistics
• Conditioning to reduce dimensionality