Coupon Collector Calculator

About this calculator

The coupon collector's problem is a classic result in probability theory: if you draw items at random (with replacement) from a set of n distinct types, how many draws do you expect to make before you've collected every type at least once? The expected number is n·Hn, where Hn is the nth harmonic number (1 + 1/2 + 1/3 + ... + 1/n), which for large n is well approximated by n·ln(n) + γn.

The counterintuitive part is that collecting the last few missing items takes disproportionately long — getting the final type alone takes n draws on average. Statisticians and computer scientists use this formula to analyze things like hash table collisions and randomized algorithms, while it also explains real-world experiences like collecting every sticker in an album or every character skin in a loot-box game.