Coupon Collector Calculator

How it is calculated

For n equally likely item types drawn at random with replacement, the expected number of draws to see every type at least once is E = n × Hn, where Hn = 1 + 1/2 + 1/3 + … + 1/n is the n-th harmonic number. The tool adds the fractions one by one (for very large n it switches to the asymptotic series for H_n), multiplies by n and rounds to one decimal. It also shows the approximation n × ln(n) + 0.5772 × n, where 0.5772 is the Euler–Mascheroni constant. n must be a whole number of at least 1: a decimal such as 6.9, zero or a negative number shows no result. The model assumes every type is equally likely on every draw, so uneven rarities need a different calculation.

Common mistakes

• Treating the result as a guarantee. 14.7 is an average over many tries; any single run can need far more or far fewer draws.
• Assuming all items are equally likely when they are not. Loot-box items with different drop rates take much longer than this formula says, because the rarest item dominates.
• Entering a non-integer count. The number of distinct items must be whole, so 6.9 gives no result rather than being rounded.
• Confusing draws with distinct items already held. The formula counts total draws including duplicates, not purchases of new items only.
• Using the approximation for small sets. For n = 6 it gives 14.2 against the exact 14.7, so rely on the n × H_n figure when n is small.