Hong Life CalculatorsUnderstanding the gacha probability and pity calculator — concept and principle
Gacha games sell random pulls in which a rare item or character appears with a small published probability. This calculator shows the chance of getting at least one copy within a given number of pulls, how many pulls give a 50% or 90% chance, the average number of pulls needed, and the most you can spend before a hard pity guarantee triggers.
How it is calculated
- P(at least one) = 1 − (1 − p)ⁿ, where p is the drop rate as a decimal and n the number of pulls. If a hard pity is set and n is at least the pity count, the result is 100%.
- Pulls for 50% / 90% = ln(1 − target) ÷ ln(1 − p), rounded up, but never more than the hard pity, since the item is guaranteed there.
- Expected pulls (no pity) = 1 ÷ p, or 100 ÷ rate in %.
- Expected pulls (with hard pity) = (1 − (1 − p)^pity) ÷ p, the average of the geometric distribution cut off at the pity.
- Worst-case cost = pity count × cost per pull.
The principle
If every pull is independent with the same chance p, the only way to get nothing in n pulls is to miss every time, which has probability (1 − p)ⁿ. Everything else contains at least one hit, hence 1 − (1 − p)ⁿ. Solving that expression for n gives the pulls needed for any target probability. The number of pulls until the first hit follows a geometric distribution, whose mean is 1 ÷ p. A hard pity caps this distribution: no player can need more than the pity count.
Worked example
- Inputs: drop rate 0.6%, 90 pulls, hard pity 90, cost per pull $1.90.
- Because 90 pulls reach the hard pity of 90, P(at least one) = 100%. Without pity it would be 1 − 0.994⁹⁰ ≈ 41.8%.
- Pulls for 50%: ln 0.5 ÷ ln 0.994 ≈ 115.2 and for 90% ≈ 382.6, but both exceed the pity, so the result is 90 / 90.
- Expected pulls without pity = 100 ÷ 0.6 = 166.67; with the pity of 90, (1 − 0.994⁹⁰) ÷ 0.006 = 69.70.
- Worst-case cost = 90 × $1.90 = $171.00.
| Pulls | P(at least one) |
|---|---|
| 22.5 | 12.66% |
| 45 | 23.72% |
| 67.5 | 33.38% |
| 90 | 100% |
| 112 | 100% |
| 135 | 100% |
| 180 | 100% |
How to read the result
When the 50% and 90% lines both equal the pity, pity rather than luck usually decides when you get the item; the 58% of players who miss in 90 pulls all get it at pull 90, so the average falls to about 70. Many games also have soft pity, where the rate rises sharply in the last stretch before hard pity, so real average pulls are lower than 1 ÷ p. A 50/50 rule on the featured item can double the worst case.
Common mistakes
- Adding probabilities: 0.6% × 90 = 54% is wrong; the correct no-pity value is 41.8%.
- Believing you are "due" after many misses; with independent pulls past failures do not raise the base rate.
Frequently asked questions
Is 1 ÷ p the number of pulls that guarantees the item?
No. It is the average. At exactly 1 ÷ p pulls the chance is only about 63%.
Are gacha rates published?
In South Korea game companies must disclose probability-item rates under the Game Industry Promotion Act, in force since March 2024.
Sources: Sheldon M. Ross, A First Course in Probability (geometric distribution) · Game Industry Promotion Act of the Republic of Korea, probability-item disclosure provisions (effective March 2024)