Coupon Collector Calculator (Expected Draws to Collect All)

Compute the expected number of draws needed to collect all n distinct items using the harmonic number.

Input

Estimate the expected number of draws needed to collect every kind. Enter the number of kinds.

Number of kinds n

kinds

Probabilities are not equal (specify weights)

Expected draws to collect all

29draws

Standard deviation

11 draws

Median estimate

30 draws

Harmonic number H_n

2.93

Calculation steps

There are n = 10 kinds. Each draw yields one kind at random.

Compute the harmonic number H_n = 1 + 1/2 + ... + 1/n, giving H_10 of about 2.93.

The expected draws are n times H_n = 10 times 2.93, about 29 draws.

The coupon collector problem asks how many random draws are needed to obtain every one of n distinct items, when each draw yields one item at random.
When every item is equally likely, the expected number of draws to collect all of them is n times H_n, where H_n is the harmonic number 1 + 1/2 + 1/3 + ... + 1/n.
For example, with 10 kinds, H_10 is about 2.93, so the expected count is roughly 29 draws. As the number of kinds grows, the last few become hard to get and the total rises sharply.
The standard deviation measures spread and comes from Var = n^2 times the sum of 1/k^2 minus n times H_n. The median estimate uses the approximation n times (ln n + ln 2).
For gacha or card collecting, the expected count is only an average. In practice you may finish in fewer draws or far more, so use the standard deviation to gauge the range.
When item probabilities are not equal, switch to the unequal probability mode. Entering weights normalizes them to sum to one and approximates the expected count by numerical integration.
If one rare item is far less likely than the rest, the expected count is dominated by the wait to draw that rare item. This helps estimate how hard a full set is to complete.

Tell us what you think of this calculator.