Door Game Expected Cost

This quant interview question is about modeling a simple game show as a discrete random variable problem and computing an expected monetary outcome. It lives at the intersection of basic probability, expectation, and fairness in games, a classic theme in quant prep and quant interviews. The setup is simple, but it encodes a random stopping time and a cost structure that must be balanced against the payoff.

It trains your understanding of discrete random variables, especially the distribution of the trial on which a unique success occurs when sampling without replacement. It also reinforces calculating expectations and linking them to a fair game condition, which is central to pricing, risk-neutral valuation, and structuring payoff profiles.

This matters for quant interviews because interviewers want to see if you can quickly turn a verbal description into a clear probabilistic model and reason about expected value under uncertainty. It reflects the kind of probabilistic thinking used in trading, market making, and risk management, and is exactly the style of question that strong quant prep should cover.

What it tests

When sampling without replacement from a finite set where only one item is 'special', the number of draws until the special item appears is uniformly distributed over the possible positions. This is because each position is equally likely to contain the special item, so the probability that the prize is found on the $k$th attempt is $1/n$ for $n$ items. The expected value of a discrete uniform distribution on $1$ to $n$ is $(n+1)/2$. This uniformity arises because the process of random selection without replacement ensures that every order of discovery is equally probable, so the waiting time to the first success is not memoryless (unlike the geometric distribution), but evenly spread across all possible positions.