Darts in the Bullseye

This quant interview question is about modeling continuous random variables over a geometric region and turning a visual dartboard setup into a precise probability calculation. It forces you to translate a casual description of a game into the language of areas, densities, and events on a continuous space, a classic theme in quant prep for trading and research interviews.

It trains your grasp of uniform distributions over two-dimensional regions, event complements, and independence across multiple trials. You must recognize how a subregion's measure determines likelihood, and how repeated, identically distributed throws combine into a single clean probability statement.

This matters for quant interviews because many market microstructure, risk, and Monte Carlo questions reduce to the same structure: independent continuous outcomes, geometric or state-space regions, and computing the chance of hitting a critical zone at least once.

What it tests

When dealing with independent random events distributed uniformly over a geometric region, the probability that an event falls within a subregion is proportional to the ratio of the subregion's measure (such as area or volume) to the total region's measure. For multiple independent trials, the probability that none fall in the subregion is the product of their individual probabilities of missing it, and the probability that at least one falls in the subregion is the complement of that product. This structure arises because independence allows us to multiply probabilities, and uniformity ensures the ratio of measures directly gives the probability. The complement rule is powerful here because 'at least one' is often easier to compute by subtracting the probability of 'none' from one.