Chocolate Chip Coverage Puzzle

This puzzle considers a random allocation of many small pieces into a fixed number of compartments, framed as chocolate chips landing in cookies after thorough mixing. The candidate must reason about how likely it is that no compartment is left completely empty, and determine how many pieces are needed to make the "no empty spot" event sufficiently likely. The core challenge is understanding how a global requirement on all compartments arises from many local random placement events, and how this probability changes as the number of pieces increases.

Solving it leans heavily on counting arguments and exact probability distributions for occupancy problems. Strong answers may use inclusion–exclusion, connections to the coupon collector or occupancy models, or suitable approximations to handle large parameters. Interviewers look for clear formulation of the underlying sample space, correct handling of dependencies between compartments, and a systematic strategy for turning a qualitative requirement ("high probability of full coverage") into a quantitative threshold. Good candidates also justify any approximations and check that their final threshold behaves sensibly.

What it tests

When distributing indistinguishable items (like `N chips`) into distinguishable bins (like `100 cookies`) at random, the probability that every bin receives at least one item is governed by the inclusion-exclusion principle. This principle systematically corrects for overcounting by alternating between adding and subtracting the probabilities of increasingly overlapping 'bad' events (such as bins being empty). The structure arises because the events of different bins being empty are not independent: the more bins you require to be empty, the less likely it is, but their overlaps must be carefully balanced. The formula thus sums over all possible subsets of bins, weighting each by the number of ways to choose that subset and the probability all chips miss it. This approach generalizes to any problem where you want to avoid empty bins, regardless of the context or the number of items and bins.