Gacha Pity Probability Calculator
Gacha games feature random draws with a pity system: if you have not received the featured item after a set number of pulls, it is guaranteed on the next pull. This calculator computes your cumulative probability of getting the featured item in a given number of pulls, accounting for both the base drop rate on each pull and the hard pity guarantee. Enter the base rate, the pity cap (guaranteed at this pull count), and how many pulls you plan to make.
Gacha pity probability formula
If n < pity cap: P(at least one in n) = 1 - (1 - p)^n
If n >= pity cap: P = 100% (guaranteed)
Expected pulls = sum k * P(first success on pull k), k=1..pity
Each pull before the pity cap is an independent Bernoulli trial with probability p. At the pity cap the item is guaranteed, capping the maximum pulls needed.
Pity systems and player experience
- A 0.6% base rate with 90-pull pity (common in many gacha games) gives about 41.5% probability at 90 pulls before pity, but pity ensures you always get the item by pull 90.
- Pity reduces the worst-case spending significantly compared to pure random drops.
- Designers set pity caps to balance monetization with player goodwill and regulatory considerations around loot boxes.
- Expected pulls with pity is always less than 1/p (the no-pity mean), since pity prevents the very long tail.
Gacha pity: frequently asked questions
What is a gacha pity system?
A pity system guarantees a featured item if the player has not received it within a set number of pulls (the pity cap). Before the pity cap is reached, each pull has the base drop rate. At the pity cap, the item is guaranteed (100% probability).
How is cumulative probability calculated with pity?
For pulls 1 through pity cap minus 1, each pull is an independent Bernoulli trial with the base probability. The cumulative probability after n pulls (where n is less than pity cap) is 1 minus (1 minus p)^n. At pull equal to pity cap, probability reaches 100%.
What is the expected number of pulls to get the item?
Without pity: 1/p. With pity at cap P: it is the average of 1/p and P, weighted by the probability of hitting pity. This calculator shows the expected pulls directly.
How does soft pity differ from hard pity?
Hard pity is a guaranteed item at the cap. Soft pity increases the drop rate as pulls approach the cap, reducing the average pulls needed without a hard guarantee. This calculator models hard pity only.
Does the pity counter reset after I get the featured item?
In most gacha games, yes. Once you receive the featured item, the pity counter resets to zero. For multi-item pity systems (e.g. guaranteed featured on the second copy), model each cycle separately.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: itl.nist.gov/div898/handbook/.
- National Council of Teachers of Mathematics: nctm.org.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.