Coupon Collector, explained.
The coupon collector problem asks how many random draws it takes to obtain every type in a collection, including repeated draws of types already collected.
Why it happens
Early discoveries are easy because almost everything is new. Near completion, most draws are duplicates. With equally likely types, collecting all n types takes n times the harmonic sum on average, rather than merely n draws.
Collecting every equally likely type has a long tail because duplicates become common near completion.
Read the result
Watch the pace slow near the final missing type. Compare many collections: an average completion time does not promise that your particular collection will finish by that point.
A worked example
Four equally likely stickers
Each packet contains one of four sticker types, with equal probability.
The expected time to complete the set is 4 × (1 + 1/2 + 1/3 + 1/4), about 8.33 packets.
Once only one type is missing, each new packet has just a one-in-four chance of completing the set.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Independent draws are uniform across the selected types. Exact expected draws to collect k types equal N times the sum of 1/(N-i) for i from zero to k-1. A seeded run shows one possible path.
Where this idea is useful
A practical use
Planning how many random packs may be needed to complete a collection.
A common misconception
“Four types should take about four packets.”
That would require avoiding duplicates. Random sampling with replacement repeatedly spends draws on types you already have.
What this explanation leaves out
- Real items may have different rarity and draws may not be independent.
What if one collectible is rare?
Unequal probabilities change the expectation. A rare final type can dominate the waiting time, so the equal-probability formula no longer applies.
Which missing item could become the bottleneck in completing your collection or coverage?
Further reading
Explore the original research or the teaching reference behind this experiment.