The formula
P(at least one) = 1 − (1 − p) to the power of the number of pulls
Where it comes from
Each pull is independent of the ones before: the system keeps no memory of what you have already spent, unless the game has an explicit pity counter. That is why the sum is done through the complement, multiplying the chance of missing as many times as there are pulls. And it is why you never reach 100 %, however many pulls you make.
How to work it out by hand
- Subtract the probability from one to get the chance of missing
- Raise it to the number of pulls: that is the chance of missing every one
- Subtract it from one
What is worth knowing
The most counter-intuitive result is that at a rate of 1/n it takes considerably more than n pulls to have a reasonable chance: at 1 %, a hundred pulls give 63 % and it takes 459 to reach 99 %. Many games offset this with a pity system that guarantees the prize after a set number of pulls, and that mechanism changes the calculation completely — it is no longer the one on this page. It is worth looking at the published effective rate, which already folds pity in, rather than the base rate.