Coupon Collector Calculator
Enter how many distinct items exist (like coupon types, cards, or prizes) to find the expected number of random draws to collect them all.
Last updated: September 5, 2026
Frequently asked questions
What is the coupon collector's problem?
A classic probability problem: if there are n distinct items and each random draw is equally likely to be any one of them (with replacement), how many draws on average does it take to collect at least one of every item?
What's the expected number of draws to collect all 6 items?
About 14.7 draws - using expected draws = n × Hₙ = 6 × (1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6) = 6 × 2.45 ≈ 14.7.
Why does it get disproportionately harder to collect the last few items?
Early draws are likely to be new items since most of the set is still uncollected, but as you collect more, each new draw has a shrinking chance of being one of the few remaining types - the last item alone takes n draws on average to find.
Does this assume every item is equally likely to be drawn?
Yes - this calculation assumes a uniform distribution, where each of the n items has an equal 1/n chance on every draw. If some items are rarer than others, the actual expected number of draws will be higher than this estimate.
How does the expected draws change as n grows larger?
It grows roughly as n × ln(n), so doubling n more than doubles the expected number of draws - for 100 items, expected draws are about 519, far more than double the ~14.7 draws needed for 6 items.
What's the minimum possible number of draws to collect all n items?
n draws - that's the theoretical best case where every single draw happens to be a new, previously uncollected item. The expected value calculated here is always higher than this minimum, since collisions with already-collected items are likely.
Is this the same math used for calculating trading card or loot box collection odds?
Yes - the coupon collector's problem is the standard model for how many random pulls it takes to complete a full set, whether that's cereal box prizes, trading cards, or in-game loot drops with equal drop rates.