Fair cost per door split

This quant interview question is about pricing a fair game when you stop as soon as you hit a winning outcome. It sits at the intersection of expected value, discrete probability, and stopping rules, and captures the essence of risk-neutral valuation. You must relate the game's fairness condition to the probabilistic structure of multiple winning outcomes hidden among indistinguishable choices.

It trains your understanding of expected value in a sequential search, conditional on success occurring eventually, with multiple rewards and no replacement. It pushes you to reason about the distribution of failures before the first success in a symmetric setup, and to translate that expectation into a fair cost per action in a repeated random experiment.

This matters for quant interviews because quant roles constantly involve pricing, expected PnL, and designing fair or neutral structures. Mastering this kind of question signals solid quant prep, comfort with probabilistic modeling, and the ability to turn qualitative game descriptions into precise mathematical expectations, a core skill in trading, risk, and derivatives research.

What it tests

This problem class is governed by the principle of expected value in sequential search with multiple successes. When searching for one of several 'winning' outcomes among a set of indistinguishable options, the expected number of trials to the first success depends on both the number of successes and their distribution among the failures. The key is that the empty spots (failures) are distributed among the intervals created by the successes, and the expected number of failures before the first success is the total number of failures divided by the number of intervals (which is one more than the number of successes). This arises because each interval is equally likely to be the one you encounter first, so the expected value is evenly shared. The pattern holds for any problem where you are searching for the first of several identical targets in a random order without replacement, and the cost or reward is tied to the expected number of steps taken.