Average Letters Delivered Correctly

This classic quant interview question is about randomness in matching and allocation. You randomly assign multiple labeled items to labeled destinations and want to know, on average, how many end up where they belong. It sits at the intersection of expected value, permutations, and symmetry, and it appears frequently in quant prep material because it is short, clean, and surprisingly general.

It trains your ability to translate a verbal story into random variables, to define appropriate indicators, and to compute an expectation without touching complicated distributions. It reinforces comfort with additivity of expectation in situations where outcomes are clearly dependent, a key habit in fast mental math during interviews.

This matters for quant interviews because it mirrors real problems in matching, allocation, and assignment in trading systems. It tests whether you recognize structure quickly, avoid unnecessary casework, and use robust probabilistic tools under time pressure.

What it tests

The core structure here is the use of indicator variables and the linearity of expectation, which allows us to compute the expected value of a sum by summing the expected values of its parts, regardless of any dependencies between them. This principle is powerful because it sidesteps the need to analyze the joint distribution or complex correlations among the variables. The symmetry in random assignments ensures that each individual event (a letter reaching its correct destination) has the same probability, so the expected total is simply the sum of these identical probabilities. This approach works broadly for counting expected occurrences of specific outcomes in random arrangements, such as fixed points in permutations or matches in random assignments. The reason this holds is that expectation is additive even when the events are not independent, making it a robust tool for problems involving random allocation or matching.