Random convex hull

What this convex hull probability question tests

This is a medium-difficulty geometry and probability problem that combines spatial reasoning with combinatorial calculation. It appears regularly in quant interviews because it requires candidates to translate a geometric intuition into a precise probabilistic argument rather than relying on brute-force simulation.

The core challenge is recognizing that the position of the center relative to a random set of points has a clean combinatorial structure. Solving it demands identifying the symmetry in the problem, understanding what configuration guarantees the center lies inside the convex hull, and then counting or weighting the relevant outcomes. The calculation itself is tractable but the setup—knowing which symmetries to exploit—is where most candidates stumble.

• Convex position and point-in-polygon tests
• Symmetry and combinatorial probability
• Polar or angular decompositions of random point sets