Dorsia Dinner: 10-Minute Overlap

This Dorsia dinner scenario is a classic continuous random variables question that shows up in top quant interviews. It frames arrival times as independent uniform choices on an interval and focuses on when those choices lead to an overlap. The setup is simple and intuitive, but the underlying probability structure is fully continuous and geometric, not discrete.

Working through this kind of quant prep trains your intuition for joint distributions, geometric probability, and symmetry in the plane. You practice turning a verbal description into a region in a two-dimensional space of possible outcomes, then relating that region to a probability. It also reinforces comfort with independence and translating real-world timing into random variables.

This matters in quant interviews because many trading and risk problems reduce to continuous timing, overlaps, and thresholds. Interviewers use this question to see whether you can quickly build a mathematical model from a short story-like description, navigate continuous probability without formulas being handed to you, and reason visually about joint distributions. It is a compact test of modeling skill, geometric thinking, and the kind of probabilistic reflexes high-frequency and derivatives firms expect.

What it tests

When two independent random events occur within the same interval and each event occupies a fixed duration, the probability that their intervals overlap is governed by the geometry of their possible start times. This is a problem of uniform probability over a square, where the event of interest corresponds to a region defined by an inequality (here, $|X - Y| \leq t$ for some $t$). The key is that the overlap condition translates to a band along the diagonal of the square, and the probability is the ratio of the area of this band to the total area. This geometric approach applies whenever independent uniform random variables are constrained by a symmetric difference or interval condition. The reason this works is that uniform distributions allow us to interpret probabilities as ratios of areas, and the overlap constraint carves out a symmetric region around the diagonal.