Sock Pair Count Expected

This quant interview question is about modeling a simple random selection process and tracking how many structured objects survive it. You work with a finite sample space and must reason carefully about which configurations are counted as successes. Although the story is casual, the underlying combinatorics and probability reasoning are exactly what shows up in real quant prep for top trading firm interviews.

It trains your understanding of expected value, indicator variables, dependency, and counting arguments. You practice turning an informal description into random variables and events, then computing expectations with clean probabilistic thinking. This is fundamental for quant interviews, where you constantly need to translate a narrative into precise mathematical objects.

It matters because front-office quant roles demand fluency with probabilistic modeling under combinatorial complexity. Interviewers use this type of question to see if you can manage dependencies without getting lost, reason about expectations efficiently, and communicate a tight, rigorous argument under time pressure.

What it tests

This problem class is governed by the principle of indicator random variables and linearity of expectation. When you want the expected count of a certain structure (like complete pairs) remaining after a random process, you can define an indicator for each structure and sum their expectations. The linearity of expectation holds regardless of dependencies between indicators, so you only need to compute the probability that one specific structure survives. This probability is often found by counting favorable outcomes for that structure and dividing by the total number of possible outcomes, leveraging combinatorial reasoning. The key is that the expectation of a sum is the sum of expectations, even when the events are not independent, which simplifies what could otherwise be a very complex dependency problem.