Gacha Pity Calculator
Calculate the probability of getting a 5-star in gacha games within N pulls including pity.
Gacha Pity Calculator
Calculator
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Formula: P(success) = 1 - Product(1 - rate_i) for i = 1 to N
Worked example — 91.8% chance of 5-star within 80 pulls. Expected pulls: ~62.
Formula
P(success) = 1 - Product(1 - rate_i) for i = 1 to N
The cumulative probability is calculated by finding the probability of NOT getting the item on every single pull, multiplying them together, and subtracting from 1. After soft pity begins, the per-pull rate increases linearly, dramatically boosting cumulative probability.
Worked Examples
Example 1: Genshin Impact Standard Pity
Problem:A player has done 80 pulls on the character banner with 0.6% base rate, soft pity at 74, hard pity at 90, and 6% increase per pull after soft pity. What is the probability of having a 5-star?
Solution:Pulls 1-73: rate = 0.6% each Pull 74: rate = 0.6% + 1 x 6% = 6.6% Pull 75: rate = 0.6% + 2 x 6% = 12.6% ... Pull 80: rate = 0.6% + 7 x 6% = 42.6% P(no 5-star in 73 pulls) = (0.994)^73 = 0.643 P(no 5-star in pulls 74-80) = 0.643 x 0.934 x 0.874 x ... x 0.574 = 0.082 Cumulative probability = 1 - 0.082 = 91.8%
Result:91.8% chance of 5-star within 80 pulls. Expected pulls: ~62.
Example 2: Budget Planning for a Guaranteed Featured Character
Problem:A player wants to know how many pulls a specific featured character can take, and how to turn that pull count into money using their own cost per pull (C).
Solution:Worst case: lose the 50/50, then the guarantee = 180 pulls = 180 x C Expected case: 1.5 five-stars needed = ~93 pulls = 93 x C Best case (win the 50/50 early): ~40 pulls = 40 x C Monthly pass income: 90 primogems/day x 30 days = 2,700 primogems 2,700 / 160 primogems per pull = ~17 pulls per month Saving for 3 months = ~51 free pulls Remaining at the expected case: 93 - 51 = 42 pulls = 42 x C
Result:~93 pulls expected, ~42 after three months of pass income, 180 worst case. Multiply by your own cost per pull.
Frequently Asked Questions
What is the pity system in gacha games and how does it work?
The pity system is a mechanic in gacha games designed to guarantee players eventually receive rare items after a certain number of unsuccessful pulls. There are two main types of pity. Soft pity begins increasing the drop rate after a certain number of pulls without a rare item, making each subsequent pull more likely to yield the desired reward. Hard pity is the absolute guarantee point where the player is certain to receive the rare item. For example, in Genshin Impact, the base 5-star rate is 0.6 percent, soft pity begins at pull 74 with rates increasing by approximately 6 percent per pull, and hard pity guarantees a 5-star at pull 90. This system prevents infinite bad luck while still maintaining the excitement of random pulls.
How do you calculate the probability of getting a 5-star within N pulls?
The cumulative probability of obtaining a 5-star within N pulls is calculated using the complement method. For each pull, you calculate the probability of NOT getting the item, multiply all those probabilities together, and subtract from 1. Before soft pity, each pull has the base rate (say 0.6 percent), so the probability of missing is 99.4 percent. After soft pity begins, the rate increases each pull. The formula is: P(success within N) = 1 - product of (1 - rate_i) for all pulls from 1 to N. For example, with a 0.6 percent base rate and no pity, 50 pulls gives approximately 1 - (0.994)^50 = 25.9 percent chance. With soft pity starting at pull 74, the cumulative probability increases dramatically between pulls 74 and 90.
What is the expected number of pulls to get a 5-star character?
The expected number of pulls depends on the specific game mechanics. In Genshin Impact with standard pity, the expected pulls for any 5-star is approximately 62 to 63 pulls on average, well below the 90-pull hard pity. This is because soft pity dramatically increases rates starting at pull 74. Without any pity system at a flat 0.6 percent rate, the expected pulls would be approximately 167 (calculated as 1/0.006). The soft pity system effectively cuts the expected pulls by more than half. For a specific featured character with the 50/50 system you need 1.5 five-stars on average, so the expectation rises by half to roughly 93 pulls, since a lost 50/50 sends you through the pity cycle a second time.
How much money does it typically cost to guarantee a 5-star in gacha games?
There is no single dependable dollar figure, so work it out in two steps rather than trusting a quoted average. First fix the pull counts, which are game mechanics and do not move with your wallet: in Genshin Impact one pull costs 160 primogems, hard pity guarantees a 5-star by pull 90, the expected number of pulls for any 5-star is around 62 once soft pity is included, and a specific featured character on a 50/50 system can take up to 180 pulls. Second, derive your own cost per pull: take the price of the top-up bundle you would actually buy in your own store, divide it by the premium currency that bundle grants, and multiply by the currency cost of one pull. Multiply that figure by the pull counts above to get a best case, an expected case and a worst case. Cost per pull is the volatile part of the calculation because it moves with region, currency, platform, sales tax and one-time first-purchase bonuses, which is why Gacha Pity Calculator asks you to supply it instead of assuming one. The structural difference that matters far more than price is whether a game has a hard pity at all: without a guarantee there is no bounded worst case, only a tail that can run indefinitely.
Does the pity counter reset between different banners?
Pity counter mechanics vary by game and banner type. In Genshin Impact, pity carries over between banners of the same type. For example, if you are at 50 pity on the character event banner and the banner changes to a new featured character, your pity remains at 50 on the new character event banner. However, pity does NOT carry over between different banner types. The character event banner, weapon banner, and standard banner each maintain separate pity counters. Additionally, the guaranteed featured character mechanic (losing the 50/50) also carries over between same-type banners. Understanding which banners share pity and which are independent is crucial for strategic pull planning and budgeting.
Which single pull is most likely to be the one that lands the 5-star?
Not the hard pity pull. The unconditional chance that the drop lands on pull i equals the survival probability through pull i-1 multiplied by that pull's rate, so it is a product of a shrinking term and a growing term. With a 0.6 percent base rate, soft pity at 74 and a 6-point ramp, that product peaks around pull 77 at roughly 10 to 11 percent, then falls away as almost nobody survives to pull 80 and beyond. The hard pity at 90 almost never actually fires with a ramp that steep; it exists as a ceiling, not as the typical outcome.
What is the difference between the advertised base rate and the consolidated rate?
The base rate is the flat per-pull chance outside the soft pity window, and it is the number printed on the banner details page. The consolidated or effective rate folds the pity mechanic in and equals one divided by the expected pulls. With Genshin-style settings the expected pulls come out near 62, so the consolidated rate is about 1.6 percent, roughly 2.7 times the 0.6 percent base rate. Use the base rate to model individual pulls and the consolidated rate when converting a primogem budget into an expected number of 5-stars.
How much of the work is done by soft pity versus hard pity alone?
Set the soft pity increase to zero and the model becomes a geometric distribution truncated by the guarantee, with expected pulls equal to (1 - (1-p)^H) / p. At p = 0.6 percent and a hard pity of 90 that gives about 70 pulls, compared with roughly 167 pulls for a flat 0.6 percent rate with no guarantee at all. Adding a 6-point soft pity ramp from pull 74 pulls the expectation down further to about 62. So the guarantee ceiling does most of the heavy lifting, and soft pity mainly tightens the spread of outcomes.
How do you estimate the soft pity ramp for a game that never publishes its curve?
Treat the ramp as a fitted parameter rather than a known constant. Take a large community pull dataset, record the pull number on which each 5-star landed, and find the histogram peak and the average. Then adjust the soft pity start and the per-pull increase in Gacha Pity Calculator until the model reproduces both. The histogram peak pins the soft pity start, and the sharpness of the drop-off after that peak pins the increase. A steep ramp produces a narrow spike just after soft pity; a shallow ramp spreads outcomes toward hard pity.
Why is the expected pull count lower than the pull count you should actually budget for?
Because the distribution is not symmetric, so the mean and the median disagree. With the default settings only about 36 percent of accounts hit before soft pity begins, and those early hits are thinly spread across 73 pulls, which drags the average down to roughly 62. The cumulative probability, though, does not cross 50 percent until around pull 76. Budget to the median or to the hard pity ceiling if you need the character, and use the mean only when averaging cost across many banners.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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