Optimal Staffing to Maximize Non-Working Hours

This quant interview question is built around birthdays and work schedules, but its real core is expected value under independence assumptions. Candidates must translate an unusual workplace story into a clean probabilistic model with a clear objective function. The setting is stylized on purpose to test whether you can strip away narrative noise and see a pure quant problem hiding underneath.

It trains your intuition for how probabilities behave as you scale the number of independent "trials," and how that interacts with linear growth in payoff. You practice setting up expectations precisely, understanding saturation effects, and spotting when an expression diverges instead of having a finite maximum. This is classic quant prep material: turning words into a sharp stochastic optimization problem.

For quant interviews, this matters because many trading and risk problems have exactly this flavor. You weigh more positions, more signals, or more bets against changing probabilities of a favorable outcome. Interviewers want to see whether you can recognize when a model suggests pushing size indefinitely, and more importantly, whether you can clearly justify that conclusion from the structure of the expectation rather than by guesswork.

What it tests

This problem class is governed by the principle of maximizing an expected value that grows both with the number of trials (here, employees) and with the probability that a desired event (at least one birthday on a given day) occurs. The key structure is that as the number of independent trials increases, the probability that at least one trial triggers the event approaches 1, while the total benefit scales linearly with the number of trials. The expected value is thus a product of a rapidly saturating probability (approaching 1) and a linearly growing factor (the number of employees), which together can cause the expected value to increase without bound. This happens because the diminishing returns from the probability saturating are outpaced by the unbounded growth of the linear factor. The underlying pattern is that, in such settings, unless there is a constraint or cost that penalizes adding more trials, the optimal solution can be infinite.