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.
Calculator
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Pull-by-Pull Breakdown
Formula
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.
Last reviewed: December 2025
Worked Examples
Example 1: Genshin Impact Standard Pity
Example 2: Budget Planning for Guaranteed
Background & Theory
The Gacha Pity Calculator applies the following established principles and formulas. Statistics and probability provide the mathematical framework for drawing conclusions from data under uncertainty. The measures of central tendency describe where data cluster. The mean is the arithmetic average, computed as the sum of all values divided by the count. The median is the middle value of an ordered dataset, robust to extreme outliers. The mode is the most frequent value. Spread is quantified by variance, the average squared deviation from the mean, and by its square root, the standard deviation. For a sample, variance uses n minus one in the denominator to correct for bias in estimation. The normal distribution, defined by its mean and standard deviation, is the cornerstone of parametric statistics. Its bell-shaped probability density follows the formula f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-0.5 * ((x - mu) / sigma)^2). The empirical rule states that approximately 68 percent of observations fall within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three. A z-score standardizes a data point by subtracting the mean and dividing by the standard deviation, expressing how many standard deviations an observation lies from the mean. In hypothesis testing, the p-value is the probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true. Confidence intervals express the range within which the true population parameter falls with a specified probability, typically 95 percent. Correlation measures linear association between two variables, with Pearson's r ranging from negative one to positive one. Correlation does not imply causation. Linear regression fits a line of the form y = a + bx to minimize the sum of squared residuals. Bayes' theorem relates conditional probabilities: P(A|B) = P(B|A) * P(A) / P(B), allowing prior beliefs to be updated on new evidence. The law of large numbers guarantees that the sample mean converges to the population mean as sample size grows. The central limit theorem states that the distribution of sample means approaches normality regardless of the population distribution, provided the sample size is sufficiently large, typically 30 or more.
History
The history behind the Gacha Pity Calculator traces back through the following developments. The mathematical study of probability emerged in the 17th century from correspondence between Blaise Pascal and Pierre de Fermat in 1654. Their exchange, prompted by a gambling problem posed by the Chevalier de Mere, established the foundations of probability theory by calculating expected outcomes through systematic enumeration of cases. Jacob Bernoulli formalized the law of large numbers in his posthumously published Ars Conjectandi of 1713, proving rigorously that empirical frequencies converge to theoretical probabilities with increasing observations. His work laid the groundwork for inferential statistics by connecting mathematical probability to observed data. Carl Friedrich Gauss developed the method of least squares around 1795 while adjusting astronomical observations, and he recognized the bell-shaped error distribution that now bears his name. Pierre-Simon Laplace independently worked on the normal distribution and proved an early version of the central limit theorem around 1810, demonstrating why errors in measurement tend toward normality. The late 19th century saw statistics emerge as a distinct scientific discipline. Francis Galton introduced regression and correlation in the 1880s while studying heredity. Karl Pearson formalized these concepts, developed the chi-squared test, and founded the journal Biometrika in 1901, establishing statistics as a rigorous academic field. Ronald Fisher transformed statistical practice in the early 20th century. His 1925 book Statistical Methods for Research Workers introduced significance testing, analysis of variance, and the concept of the p-value as a decision threshold, establishing the framework still used in scientific research. Fisher and Jerzy Neyman engaged in a prolonged methodological dispute over the interpretation of hypothesis tests. The Bayesian approach, rooted in the 18th century work of Thomas Bayes and Laplace, was largely eclipsed by frequentist methods through much of the 20th century but experienced a revival after World War II and accelerated with computational advances. The late 20th and early 21st centuries brought statistics into every domain through big data, machine learning, and the routine availability of software capable of processing millions of observations.
Frequently Asked Questions
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\nPull 74: rate = 0.6% + 1 x 6% = 6.6%\nPull 75: rate = 0.6% + 2 x 6% = 12.6%\n...\nPull 80: rate = 0.6% + 7 x 6% = 42.6%\nP(no 5-star in 73 pulls) = (0.994)^73 = 0.643\nP(no 5-star in pulls 74-80) = 0.643 x 0.934 x 0.874 x ... x 0.574 = 0.052\nCumulative 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\nExpected case: ~105 pulls = $168\nBest case (win 50/50, early): ~40 pulls = $64\nWith Welkin Moon ($5/month for 90 primos/day):\n30 days x 90 = 2,700 primos = ~17 pulls/month\nSaving for 3 months = ~51 free pulls\nRemaining needed (expected): 54 pulls = ~$86
Result: Expected cost: $168. With monthly pass savings: ~$86. Worst case: $288.
Frequently Asked Questions
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.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
How do I verify Gacha Pity Calculator's result independently?
The Formula section on this page shows the equation used. You can reproduce the calculation manually or in a spreadsheet using those steps. Compare your answer against the worked examples in the Examples section, which use known reference values so you can confirm the calculator is behaving as expected.
What inputs do I need to use Gacha Pity Calculator accurately?
Each field is labelled with the required unit (metric or imperial). Gather your source values before starting — for example, a weight measurement in kilograms, a distance in metres, or a dollar amount — and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
Why might my result differ from another tool or reference?
Differences typically arise from rounding conventions, the specific version of a formula (for example, simple vs compound interest), or unit inconsistencies between inputs. Check that both tools are using the same formula variant and the same units. The References section links to the authoritative source behind the formula used here.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy