What this calculator does
Collecting a set of random drops is not a matter of patience divided evenly. The first few items arrive quickly, because almost anything is new. The last one takes as long as everything before it put together, because only one card in the deck still helps you.
This is the coupon collector problem, and it has an exact answer. For a set of 50 equally likely items you need about 225 drops, of which 175 are duplicates. The final item alone accounts for 50 of those drops, more than a fifth of the whole grind, and no amount of skill changes that.
The formula
When you hold k of N distinct items, the chance that the next drop is new is (N − k) ÷ N, so you wait N ÷ (N − k) drops on average for it. Adding those waits from your current position gives N × (1/1 + 1/2 + … + 1/(N − k)), a harmonic sum. If only some attempts drop anything at all, divide the result by that chance to turn drops into attempts.
| Term | Meaning |
|---|---|
| N | How many distinct items make up the complete set. |
| k | How many distinct ones you already hold. Duplicates do not count. |
| Harmonic sum | 1 + 1/2 + 1/3 + … The reason the total grows a little faster than N itself. |
| Drop chance | The probability an attempt yields any item at all, as opposed to nothing. |
The inputs explained
| Field | What to enter |
|---|---|
| Distinct items in the set | The size of the full set. The calculation assumes every item is equally likely, which is the usual design. |
| Distinct items already owned | Distinct items owned, not total items. Having six copies of one thing counts as one. |
| Chance an attempt yields an item (%) | Leave at 100% if every attempt gives you something. Lower it if most attempts come back empty. |
When to use it
Deciding whether to finish a set
Enter what you already have. Going from 45 of 50 to the full set takes about 114 more drops, which is half the cost of the entire set from scratch. That is the point where many people quietly stop, and the arithmetic says they are not being unreasonable.
Comparing a set against a single target
A specific item at a 2% rate averages 50 attempts. A 50-item set at guaranteed drops averages 225. Set completion is usually the longer commitment even though each individual item is common.
Budgeting a seasonal event
If an event runs for a fixed number of attempts, compare that to the expected total. Expected does not mean guaranteed, and roughly half of players will need more than the figure shown.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How many attempts does a full set take?
Only the size of the set changes.
| Items in the set | Attempts still needed | Drops spent on the final item alone | Share of the grind that is the last item |
|---|---|---|---|
| 10 | 29 | 10 | 34.1% |
| 20 | 72 | 20 | 27.8% |
| 30 | 120 | 30 | 25.0% |
| 50 | 225 | 50 | 22.2% |
| 100 | 519 | 100 | 19.3% |
| 200 | 1,176 | 200 | 17.0% |
Why is the last item so slow?
The same set, picked up from different stages of completion.
| Distinct items owned | Attempts still needed | Drops still needed |
|---|---|---|
| 0 | 225 | 225.0 |
| 25 | 191 | 190.8 |
| 40 | 146 | 146.4 |
| 45 | 114 | 114.2 |
| 48 | 75 | 75.0 |
| 49 | 50 | 50.0 |
Questions
Does this assume every item is equally likely?
Yes. Rarity tiers break the assumption and make the real figure larger, sometimes much larger, because the rare items dominate the wait. Treat the result as a floor if your set has rarities.
Is the expected number the number I should plan for?
No. It is the long-run average, and the distribution has a long tail, so a substantial share of players will need noticeably more. It is a reasonable central estimate, not a budget.
Why do duplicates matter so much?
Because once you hold most of the set, almost every drop is a duplicate by definition. At 49 of 50 items, 98 per cent of drops are wasted, which is exactly why the last one takes 50 attempts.
What if duplicates convert into currency?
Then the real cost is lower than this suggests, and the conversion rate matters more than the drop rate. Work out how many duplicates buy a chosen item and compare that against the expected duplicate count shown here.
How does this relate to a single rare drop?
A single target is a simpler problem: the expected attempts are one divided by the drop rate. The drop rate calculator handles that case, including the chance of success within a set number of tries.
For a single item rather than a set, see the drop rate calculator, and for banner pulls with a guarantee there is gacha pity. Why dry streaks are the normal case is covered in the piece on pull odds.