Coupon collector's problem

The coupon collector's problem asks how many independent draws from equally likely types are needed to collect every type. For n types, the expected number of draws is n times the nth harmonic number.

Geometric distribution: A probability distribution for the number of independent trials until the first success. Each stage waits for a draw from one of the still-missing types.
Coupon collector's problem tail bounds: Bounds on the probability that coupon collection takes much longer or shorter than its typical duration. They quantify how often actual collection times depart from the expectation.
Species accumulation curve: A curve showing how the number of distinct species observed grows with sampling effort. Random species sampling raises the same question of when all types have appeared.
Expected value: The probability-weighted average of a random variable's possible values. The problem asks for the average number of draws until complete collection.
Harmonic number: The nth harmonic number is the sum of reciprocals from 1 through n. Adding the stage-wise waiting times yields the harmonic-number expectation.
Gumbel distribution: A family of probability distributions arising as limits for suitably normalized maxima of random variables. The centered collection time approaches a Gumbel law as the number of types grows.
Occupancy problem: A family of probability problems about distributing objects among locations and counting their occupancy patterns. Coupon collection is the special occupancy question of when no type remains empty.
Linearity of expectation: The principle that the expectation of a sum equals the sum of expectations, even without independence. It lets the waiting times for successive new types be added simply.
Tail-sum formula: A method for finding a nonnegative integer-valued random variable's expectation by summing its tail probabilities. It derives the expected collection time directly from the chance that some type remains unseen.
Balls-into-bins model: A probability model that places objects independently and uniformly into a set of bins. Uncollected types correspond to empty bins after repeated random placements.

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