Optimal Group Size for Unique Birthdays

This quant interview question sits squarely in the classical occupancy framework: people are randomly assigned to equally likely buckets and you track how many buckets end up with exactly one occupant. It is a pure expected value puzzle, but with a twist: instead of just computing an expectation, you must understand how it changes as the population grows and where it peaks.

It trains comfort with modeling randomness using indicator variables, manipulating expectations, and reasoning about limits and optimization in a probabilistic setting. Strong candidates will see how to exploit symmetry and independence structures, and how to move from a discrete, combinatorial description to a clean probabilistic expression they can analyze.

This matters for quant interviews because it mirrors real quant work: interpreting a stylized model, extracting the right random variables, and analyzing non-monotone behaviors in expectations. It is ideal quant prep for roles in statistical arbitrage, risk modeling, and systematic trading.

What it tests

This problem class is governed by the principle of balancing increasing opportunities with increasing collisions. As you add more entities (people) to a fixed set of categories (birthdays), the expected number of unique categories initially rises because each new entity has a chance to occupy a new, unshared category. However, as the set fills, the probability that a new entity lands in an already-occupied category increases, and the chance of a unique assignment for any individual falls. The expected number of unique categories is maximized at the point where the marginal gain from adding another entity is exactly offset by the increased likelihood of overlap. This is a manifestation of the occupancy problem, where the expected number of singleton bins (categories with exactly one occupant) is maximized at a specific balance point between sparsity and crowding.