Gacha Pity Calculator
Calculate the probability of getting a 5-star in gacha games within N pulls including pity. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer
Gacha Pity Calculator
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Formula: P(success) = 1 - Product(1 - rate_i) for i = 1 to N
Worked example — 94.8% 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 = 0.052 Cumulative probability = 1 - 0.052 = 94.8%
Result:94.8% chance of 5-star within 80 pulls. Expected pulls: ~62.
Example 2: Budget Planning for Guaranteed
Problem:A free-to-play player wants to know the cost of guaranteeing a specific featured character assuming worst-case 180 pulls (two full pity cycles for 50/50 loss).
Solution:Worst case: 180 pulls at ~$1.60/pull = $288 Expected case: ~105 pulls = $168 Best case (win 50/50, early): ~40 pulls = $64 With Welkin Moon ($5/month for 90 primos/day): 30 days x 90 = 2,700 primos = ~17 pulls/month Saving for 3 months = ~51 free pulls Remaining needed (expected): 54 pulls = ~$86
Result:Expected cost: $168. With monthly pass savings: ~$86. Worst case: $288.
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, the expected pulls roughly doubles to about 105 pulls on average, since you have a 50 percent chance of getting the wrong 5-star and needing to go through the pity cycle again.
How much money does it typically cost to guarantee a 5-star in gacha games?
The cost varies significantly between games. In Genshin Impact, one pull costs 160 primogems, and the most efficient purchase is the Genesis Crystal top-up which works out to approximately $1.50 to $2.00 per pull depending on bonuses. At hard pity of 90 pulls, worst case cost is roughly $144 to $180. The average cost for any 5-star is around $95 to $125 given the expected 62 pulls. For a guaranteed specific featured character (accounting for the 50/50 mechanic), worst case is 180 pulls or approximately $288 to $360. Other games have different economics: Honkai Star Rail has similar pricing, while games like Fate Grand Order can require hundreds of dollars with no hard pity guarantee.
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.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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