100 Points in a Semicircle

This classic quant interview question is about random events on a geometric object, with a focus on symmetry and uniform distributions. You are asked to reason about many independent samples on a circle and understand when a global geometric condition is satisfied. It sits at the intersection of probability, geometry, and combinatorics, a staple in strong quant prep.

It trains your grasp of rotational invariance, conditional probability, and how to translate a geometric configuration into a precise probabilistic statement. You must manage independence, continuous distributions, and combinatorial counting, and then compress everything into a clean closed-form expression.

It matters for quant interviews because it mimics how you reason about correlated events, coverage, and extremal configurations in markets and risk models. Interviewers see if you can structure randomness rigorously, not just compute.

What it tests

For problems involving random points on a circle and containment within an arc (like a semicircle), the key structure is rotational symmetry and independence. The probability that all points fall within a specific arc can be analyzed by fixing one point as a reference, then considering the relative positions of the others. Because the points are independent and uniformly distributed, the chance that each subsequent point falls within a given arc is proportional to the arc's length divided by the circle's circumference. The principle holds because, by symmetry, any point could be the 'anchor' for the arc, and the events for each point are independent, so the total probability is a product of these independent probabilities, multiplied by the number of possible anchor points (to account for all possible arcs). This approach generalizes to arcs of any fixed length, not just semicircles, and is rooted in the uniformity and independence of the random placements.