Coupon Collector Calculator (How Many Packs to Complete the Set)
How many random draws you need on average to collect every different item in a set, or a given number of them, with the typical spread.
Results
The dashed line marks the value you entered.
What this tool does
If every draw is equally likely to be any of the n items, the first few new ones come quickly and the last ones very slowly: the final missing item alone takes about n draws on average. The total for a full set is n × (1 + 1/2 + … + 1/n), roughly n (ln n + 0.58): 50 items need about 225 draws and 100 items about 519, over five times the size of the set. The spread is large — it is not unusual to need well over the average — and real collections with rare items, or packs that avoid duplicates, behave differently.
Formula
E[draws] = n (1 + ½ + ⅓ + … + 1⁄n) ≈ n (ln n + 0.5772) · for k different: n (1⁄n + 1⁄(n−1) + … + 1⁄(n−k+1))
Variables
| Symbol | Meaning | Unit |
|---|---|---|
nn | Different items in the full set | — |
k | How many different ones you want (same number = whole set) | — |
E | Expected number of draws | — |
SD | Typical spread (standard deviation) | — |
EX | Duplicates you can expect along the way | — |
PI | Draws per different item collected | — |
Worked example
- Different items in the full set100
- How many different ones you want (same number = whole set)100
- Expected number of draws518.7
- Typical spread (standard deviation)125.8
- Duplicates you can expect along the way418.7
- Draws per different item collected5.19
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.