What this calculator does
A published rate of 0.6% per pull does not mean much on its own, because what you actually want to know is the chance across the forty pulls you can afford. That is one minus the chance of missing every time, and it comes out far higher than the headline rate and far lower than most people guess.
Hard pity changes the shape of it completely. If the game guarantees a copy at pull 90, then your expected number of pulls is not 1 ÷ 0.006 = 167, it is around 70, because the guarantee truncates the long tail. Any estimate that ignores pity is badly wrong for exactly the players who most want the answer.
The formula
The chance of at least one copy in n pulls is 1 − (1 − rate)^n, since each pull independently misses with probability 1 − rate. Once n reaches the hard pity count the chance is 100% by definition. The expected pulls figure sums over every outcome: the chance your first copy lands on pull 1, pull 2 and so on up to the pity pull, where whatever is left over is guaranteed. The pulls-for-a-given-chance figures invert the first formula, and are capped at the pity count.
| Term | Meaning |
|---|---|
| Rate | The published probability per pull, as a percentage. |
| Hard pity | The pull number at which a copy is guaranteed if you have not had one. |
| Expected pulls | The long-run average number of pulls to your first copy, allowing for pity. |
| Chance within n pulls | The probability of at least one copy, not of exactly one. |
The inputs explained
| Field | What to enter |
|---|---|
| Rate per pull (%) | The published rate per pull. Enter 0.6 for 0.6%, not 0.006. If your game has a soft pity where the rate climbs near the end, this calculation will be conservative. |
| Pulls you can make | How many pulls your current currency buys. |
| Hard pity at (pulls) | The guaranteed pull number. If your game has no hard pity, set this to something very large. |
When to use it
Deciding whether you can afford to pull
Enter the pulls your current stock buys and read the chance. If it is lower than you are comfortable with, the pulls-for-a-50%-chance figure tells you how much more you need to save.
Budgeting for a guarantee
The only number that guarantees anything is the hard pity count. Saving to that figure removes the gamble entirely, and the expected pulls figure tells you what you will typically spend if you do not.
Understanding why you went dry
At 0.6% per pull, going 50 pulls with nothing happens about 74% of the time. A dry streak is the normal case, not bad luck, and the chance-of-nothing column makes that concrete.
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.
What are my odds at 0.6% per pull?
The chance of at least one copy as the pull count climbs.
| Pulls | Chance within those pulls | Chance of nothing |
|---|---|---|
| 10 | 5.84% | 94.2% |
| 20 | 11.3% | 88.7% |
| 30 | 16.5% | 83.5% |
| 50 | 26.0% | 74.0% |
| 70 | 34.4% | 65.6% |
| 89 | 41.5% | 58.5% |
| 90 | 100.0% | 0.000% |
How much does the published rate change things?
Different games publish very different rates. Here the pull budget is fixed.
| Rate per pull | Chance within those pulls | Expected pulls to the first one |
|---|---|---|
| 0.6% | 21.4% | 69.7 |
| 1% | 33.1% | 59.5 |
| 1.6% | 47.5% | 47.9 |
| 3% | 70.4% | 31.2 |
| 5% | 87.1% | 19.8 |
Questions
Does this handle soft pity?
No. It assumes a flat rate until the hard pity pull. Games where the rate ramps up over the last stretch will do better than this calculation suggests, so treat the result as a floor.
What about the 50/50 on limited banners?
Not included. This gives you the chance of winning the pull itself. If your game then makes you win a coin flip for the featured character, halve the effective rate or double the pulls you budget.
Why is the expected pulls figure lower than 1 divided by the rate?
Because hard pity cuts off the long tail. Without pity at 0.6% the average would be about 167 pulls. The guarantee at 90 drags that down to around 70.
Are pulls really independent?
Between pity resets, yes, in every system that publishes its rates. Each pull has no memory of the last one, which is why a dry streak does not make the next pull any better until pity starts to apply.
For the probability of a specific item from an enemy rather than a banner, the drop rate calculator handles repeated attempts, and XP to next level covers the other kind of grind.