Green Apples Left After Harvest

This probability question is about understanding what happens, on average, when you randomly exhaust one category in a mixed collection and then stop. It appears in quant prep materials because it sits at the intersection of expected value, symmetry, and order statistics, all central themes in quantitative finance interviews. Top trading firms like to see whether candidates can translate a casual story about apples or balls and urns into a clean probabilistic model.

It trains comfort with expected value in discrete settings, reasoning about random permutations, and recognizing symmetry without brute-force enumeration. Candidates must see through the narrative to realize it is about how different types are interspersed in a random ordering and what the stopping rule implies.

This matters for quant interviews because it reflects how you reason about stopping times, conditional structure, and unbiased expectations. In trading, risk, and systematic research, you constantly analyze processes that terminate when a trigger occurs: barrier hits, defaults, liquidations, or rebalancing events. Interviewers at top trading firms use such problems to test whether you can handle random horizons, think cleanly about independence and symmetry, and explain your reasoning under time pressure.

What it tests

When objects are removed sequentially from a set containing distinguishable groups, and a process is stopped upon the last removal of a particular group, the remaining objects are those that were not yet encountered in the sequence. The key is that the stopping condition (removal of all objects of a certain type) partitions the remaining objects into segments whose expected sizes are determined by symmetry. If the order of removal is random, each possible segment (before the first, between, and after the last of the special objects) is equally likely to contain any given object from the other group. This uniformity means that, in expectation, the remaining objects are evenly distributed among the segments, and the segment after the last special object represents the expected remainder.