A gacha game tells you a 5-star has a 0.6% chance per pull. That number is real, published, and checkable. What most players never see is the second number behind it: the pity curve that quietly rewrites the odds after a long dry streak. Here is the actual math publishers use, worked through with real, cited rates so you can calculate your own odds before you spend a dollar.
The Short Answer: How Gacha Drop Rates Are Actually Calculated
Every gacha pull starts as a simple probability problem. A developer sets a base rate, say 0.6% or 0.8%, and in a pure random system each pull is an independent trial at that fixed rate. The formula for landing at least one success in N pulls is P = 1 − (1 − p)^N, where p is the base rate and N is the number of pulls. That’s the entire math behind a flat-rate system.
Almost no major gacha game stops there. Most layer a pity system on top, which turns a fixed probability into a changing one. Instead of a flat 0.6% forever, the rate holds steady for dozens of pulls, then climbs sharply, then hits a hard guarantee at a fixed pull count. That shift from constant probability to an increasing “hazard rate” is the actual mechanism behind soft pity, and it’s why the naive 1-in-166 math (1 divided by 0.6%) overstates how long a real player waits. Below, we calculate the real curve using Genshin Impact’s published Wish system as the worked example, because HoYoverse discloses more of its rate structure than most publishers.
The Core Formula Behind Every Drop Rate
For a flat-rate system with no pity, the probability of at least one success across N independent pulls is derived from the complement rule. First calculate the probability of zero successes, which is (1 − p)^N. Then subtract that from 1.
# Flat-rate cumulative probability
p = 0.006 # base rate, e.g. 0.6%
n = 73 # number of pulls
p_at_least_one = 1 - (1 - p) ** n
# p_at_least_one = 1 - 0.994^73 = 0.3556 -> 35.56%That equation only holds when every pull has the same probability, independent of what happened before. It’s accurate math, but it describes a system that barely exists in modern gacha design. The moment a game changes the odds based on how many pulls you’ve made without a hit, the pulls stop being identically distributed, and the formula needs to shift from a single fixed p to a sequence of pull-specific probabilities, p1 through pN. The cumulative formula becomes P = 1 − the product of (1 − pk) for every pull k from 1 to N. Every soft-pity game runs on this second, state-dependent version.
Why Pity Breaks the Simple Math
Soft pity means the per-pull probability is conditional on a counter, usually the number of consecutive pulls without a top-rarity hit. Below a threshold pull, the rate stays flat at the base value. Above that threshold, the rate climbs, either linearly, in steps, or on a curve set by a lookup table the developer controls internally. At a fixed pull count, “hard pity” forces the probability to 100%, guaranteeing a hit regardless of what the running total says.
This design choice is deliberate, and it’s also the reason two games with an identical base rate can feel completely different to play. A flat 0.6% rate with no pity has an expected pull count of 1 divided by 0.006, or about 167 pulls, before the first hit on average. Add a soft-pity ramp that reaches 100% by pull 90, and the real expected value drops to roughly 62 pulls, as we calculate in detail below. The base rate alone tells you almost nothing about how long you’ll actually wait.
Worked Example 1: Genshin Impact’s Rate Curve, Pull by Pull
Genshin Impact’s Character Event Wish banner publishes two hard numbers: a 0.6% base rate for a 5-star, and a guarantee that a 5-star lands by wish 90. Player-data analyses aggregated across millions of pulls, cross-referenced by outlets including BitTopup and the Genshin Impact community wikis, consistently place the start of the soft-pity ramp at wish 74, with the rate jumping to roughly 7% that pull and climbing by about 6 percentage points per pull afterward until it hits 100% at wish 90. HoYoverse has not published the exact intermediate curve, so the pull-by-pull numbers below use that widely cited approximation anchored to the two official numbers Genshin does disclose.
| Pull Number | Per-Pull Rate | Cumulative Chance of a 5-Star |
|---|---|---|
| 1–73 | 0.6% (flat) | 35.6% by pull 73 |
| 74 | ~7.0% | 40.1% |
| 76 | ~18.6% | 57.5% |
| 78 | ~30.2% | 77.6% |
| 80 | ~41.8% | 91.6% |
| 83 | ~59.2% | 99.2% |
| 86 | ~76.6% | 99.98% |
| 90 | 100% (hard pity) | 100% |
Read that table carefully and the shape of the curve becomes obvious. Through pull 73, you’re stuck at just over a one-in-three chance of ever seeing a 5-star. Six pulls later, at pull 80, your odds have jumped past 91%. The entire outcome of a pity system gets decided in a narrow 16-pull window, not spread evenly across the whole cycle. That’s also why players describe the soft-pity zone as feeling like the game “owes” them a hit, when what’s actually happening is a scheduled, disclosed acceleration in the base rate.
The Pity-Cost Table: Primogems and Real Cash to Guarantee a 5-Star
One Wish costs 160 Primogems on Genshin’s Character Event banner. That fixes every other number in the table below. Hard pity guarantees a 5-star by wish 90, which costs 14,400 Primogems if you’re only chasing any 5-star. If that 5-star isn’t the featured character, the game’s 50/50 rule guarantees the next 5-star will be, which can push the total to 180 wishes and 28,800 Primogems in the worst case. Genesis Crystals convert to Primogems 1:1, and the current standard US pack tiers run from $0.99 for 60 crystals up to $99.99 for 6,480 crystals, per official store pricing tracked by top-up guides including BitTopup’s 2026 Primogem economy breakdown.
| Scenario | Pulls | Primogems Needed | Cash Cost at Standard Pack Rates |
|---|---|---|---|
| Single 10-pull | 10 | 1,600 | ~$29.98 (two 980-crystal packs) |
| Any 5-star, hard pity | 90 | 14,400 | ~$229.97 (two 6,480 packs + one 1,980 pack) |
| Featured 5-star, worst case (lost 50/50) | 180 | 28,800 | ~$444.94 (four 6,480 packs + smaller packs to cover the remainder) |
| Featured 5-star, expected average | ~93 | ~14,900 | ~$226–$230 |
The gap between the “worst case” row and the “expected average” row is the entire point of this exercise. A player who assumes they need to budget for the full 180-pull disaster scenario is over-preparing by roughly double. A player who assumes the base 0.6% rate applies the whole way is under-preparing by nearly triple. The real number sits in between, and it comes from doing the pull-by-pull calculation rather than trusting either extreme.
Step-by-Step: Calculating the Expected Value of a Pull Sequence
Expected value for “pulls until first success” isn’t calculated by taking 1 divided by the base rate when pity is involved. Instead, it’s the sum of survival probabilities: add up the chance you’re still pulling (haven’t hit yet) after 0 pulls, after 1 pull, after 2 pulls, and so on, all the way through the pull before the hard-pity guarantee. Here’s the walkthrough using the Genshin curve from the table above.
- Step 1. Sum the survival probability across the flat-rate zone, pulls 1 through 73, using the geometric series formula. This adds up to roughly 59.9 expected pulls worth of “still waiting” probability.
- Step 2. Add the survival probability contributed by the soft-pity zone, pulls 74 through 89, using the accelerating per-pull rates from the table. This zone adds about 2.4 more expected pulls, because the probability of still needing more pulls collapses fast once the ramp kicks in.
- Step 3. Add the two together: 59.9 + 2.4 = approximately 62.3 expected pulls for any 5-star, ignoring the 50/50 rate-up mechanic.
- Step 4. Layer in the 50/50. Half the time your first 5-star is the featured character and you stop at that ~62.3-pull average. The other half, you get a non-featured 5-star and the game guarantees your very next 5-star is the featured one, which takes another ~62.3 pulls on average (the pity counter resets after every 5-star). Weighted: 62.3 + (0.5 × 62.3) ≈ 93.5 expected pulls for the featured character specifically.
- Step 5. Convert to currency: 93.5 pulls × 160 Primogems = 14,960 Primogems, which lands almost exactly on the hard-pity cost of 14,400. In practice, the 50/50 mechanic makes the expected cost for a featured unit land close to what a player would pay assuming zero luck, because the pity reset after a loss compresses the distribution.
That 93.5-pull figure matches long-running player-tracked averages for Genshin’s featured-character banner, which community trackers have historically clustered in the high-80s to mid-90s range across large sample sizes. The math and the observed data agree, which is the actual test of whether a disclosed rate is holding up in production.
Worked Example 2: What Losing the 50/50 Does to the Math
Say you hit hard pity at wish 90 and lose the coin flip, meaning your 5-star is a random standard character instead of the featured one. The math resets your pity counter to zero, but your guarantee state flips to “next 5-star is featured.” Your odds for the next 5-star follow the exact same 0.6%-to-100%-by-90 curve as before, except this time there’s no coin flip at the end, since the guarantee overrides it. That means your second cycle is a flat replay of the same 62.3-pull expected value, but guaranteed to land the character you actually want.
In the absolute worst case, both your first and second cycles run all the way to pull 90 with zero lucky hits, for 180 total pulls. Using our soft pity vs. hard pity breakdown, that 180-pull ceiling is a hard mathematical maximum, not a probability. It cannot go higher, because pull 90 forces a hit and the guarantee forces the second hit to be the featured character. Treat 180 pulls as your absolute budget ceiling and 93 pulls as your realistic planning number.
Worked Example 3: Wuthering Waves and the Limits of “Transparent” Odds
Kuro Games’ Wuthering Waves runs a comparable structure on paper: a 0.8% base rate for a 5-star Resonator, an 80-pull hard pity, and a 50/50 with a next-copy guarantee if you lose it. Community guides place soft pity starting somewhere around pull 65 to 70, but unlike Genshin, Kuro has not published enough data points, and no large-scale player tracker has published a verified pull-by-pull curve the way the Genshin community has. That gap matters for this explainer’s purpose: it’s the difference between a game where you can do the full expected-value math and one where you can only calculate the boundaries.
What we can calculate with confidence is the cost boundary. At 160 Astrite per pull, 80 pulls costs 12,800 Astrite, which runs roughly $195 to $200 using Kuro’s standard Lunite pack pricing (Lunite converts 1:1 to Astrite). Losing the 50/50 and needing a second 80-pull cycle roughly doubles that, landing near $390 to $400 as an absolute ceiling. Without a verified soft-pity curve, the true expected-value number for Wuthering Waves sits somewhere between the flat-rate math (1 ÷ 0.008 = 125 pulls, capped at 80 by hard pity) and Genshin’s compressed average, but we won’t publish an invented midpoint. That’s the anti-fabrication line: when a publisher hasn’t disclosed enough for the calculation to be verifiable, the honest answer is a bounded range, not a fabricated single figure.
Worked Example 4: NIKKE’s Flat-Rate Model, No Curve At All
Goddess of Victory: NIKKE is worth including here precisely because it does the opposite of Genshin. Its limited Recruitment banners run a flat 4% total SSR rate with no soft-pity ramp and no rate escalation at all, per current NIKKE Synergy and GachaWiki documentation. The featured character sits at a flat 2% inside that 4%. Because there’s no curve, the flat-rate formula from the top of this article applies cleanly for the entire pull sequence: expected pulls to a featured SSR equals 1 divided by 0.02, or 50 pulls on average.
Instead of a rising hazard rate, NIKKE caps the downside with a deterministic mileage system: every recruit earns one Gold Mileage Ticket, and 200 tickets can be exchanged directly for the featured character, independent of RNG entirely. That’s a fundamentally different calculation model. Genshin compresses the tail of a probability distribution with an accelerating rate. NIKKE just lets the probability run flat and backstops it with a fixed-count purchase mechanic. Both are legitimate calculable systems, they just use different math. Current published sources did not include a verified real-money Gem price table for NIKKE at the time of research, so we’re leaving the cash-cost comparison out for this one rather than guess at a figure.
| Game | Base Rate | Rate Model | Hard Guarantee |
|---|---|---|---|
| Genshin Impact | 0.6% | Flat, then accelerating curve from pull 74 | Pull 90 |
| Wuthering Waves | 0.8% | Flat, then unverified soft-pity curve near pull 65–70 | Pull 80 |
| NIKKE | 4% (2% featured) | Flat for the entire sequence, no ramp | 200-pull mileage exchange |
Worked Example 5: What a $30 Top-Up Actually Buys You
Here’s a number worth sitting with before you spend anything. A $29.99 Genesis Crystal pack buys 1,980 Primogems, which converts to 12 full Wishes with 60 Primogems left over. Twelve pulls sits well inside Genshin’s flat-rate zone, nowhere near the pull-74 soft-pity threshold. Plugging 12 pulls into the base formula: 1 − (1 − 0.006)^12 = 1 − 0.9303 = 0.0697, or roughly a 7% chance of pulling any 5-star at all, let alone the specific featured one. Our gacha pity breakeven analysis covers this same gap across other titles, and the pattern holds everywhere: a single small top-up buys you a lottery ticket, not a purchase. The pity system exists precisely to convert that lottery into a guaranteed transaction, but only once you’ve paid for enough pulls to reach it.
Who Actually Makes Publishers Show Their Math
Disclosure rules vary sharply by country, and that’s the reason some games publish detailed rate tables while others say almost nothing. China’s Ministry of Culture notice took effect in May 2017 and requires operators of randomized-item systems to disclose draw probabilities, a rule that academic compliance research published through DiGRA found is followed unevenly, with roughly 90% of studied games disclosing at least one loot box’s odds but only about 42% disclosing every relevant probability players actually encounter.
South Korea went further. An amendment to Article 33 of the Game Industry Promotion Act, passed in March 2023 and effective March 22, 2024, made probability disclosure a statutory legal duty rather than an industry courtesy, according to Chambers’ 2025 gaming law practice guide. A further amendment effective January 31, 2025 added special litigation rules and shifted the burden of proof toward game providers in disclosure disputes, and legal analysis from Kim & Chang reports that punitive-damages provisions followed in August 2025. A separate peer-reviewed compliance study found regulator-driven disclosure in South Korea already outperforming years of prior industry self-regulation. Our loot box odds disclosure roundup covers the country-by-country breakdown in more depth, including where enforcement has actually produced fines versus where the law exists mostly on paper.
Japan takes a lighter-touch approach, relying on industry self-regulation through bodies like the Computer Entertainment Supplier’s Association and the Japan Online Game Association, plus general consumer-protection law, rather than a single statute mandating every probability be published. Belgium’s Gaming Commission has treated certain paid loot boxes as unlicensed games of chance since a 2018 ruling, though enforcement has been inconsistent, a pattern also flagged in a City University of Hong Kong compliance analysis. The United States has no federal probability-disclosure law at all, and a jurisdiction-by-jurisdiction legal review confirms American gacha spending remains effectively unregulated compared to the two biggest mobile-gaming markets in Asia.
How Anyone Checks a Publisher’s Math Against Reality
There’s no global certification body that stamps a gacha game’s random number generator the way a financial auditor signs off on a balance sheet. What exists instead is statistical verification after the fact, run by players and independent trackers rather than regulators.
Binomial Testing in Plain English
If a game claims a flat 0.6% rate and a large sample of players logs their outcomes, the number of hits across n trials should follow a binomial distribution centered on n times p. A community tracker pooling, say, 500,000 logged pulls at a claimed 0.6% rate should see roughly 3,000 hits, give or take a calculable margin based on standard deviation. If the observed hit rate consistently lands far outside that margin across a large enough sample, that’s a statistical red flag worth investigating. This is exactly how community-run pity trackers for games like Genshin and Honkai: Star Rail have, over years of aggregated data, repeatedly confirmed the officially disclosed base rates and pity thresholds hold up in production.
What a Mismatch Would Look Like
A real mismatch wouldn’t show up as one unlucky player complaining online, since individual variance at low probabilities is expected and often extreme. It would show up as a persistent, large-sample deviation that survives scrutiny after outliers are filtered out, ideally paired with a configuration bug acknowledged by the developer, a compensation payout, or a regulatory finding. Current published research on 2025 and 2026 loot box compliance documents plenty of disclosure violations, meaning games not publishing required odds at all, but it does not document a widely confirmed case of a major game’s live RNG implementation provably diverging from its own published numbers. Treat that distinction carefully: “didn’t disclose” and “disclosed wrong” are different failures, and the evidence available right now points overwhelmingly to the first one.
PRNG vs. “True” Random: What’s Actually Running Under the Hood
Every pull in a gacha game is generated by a pseudo-random number generator, or PRNG, a deterministic algorithm that produces a sequence of outputs from an internal state called a seed. It isn’t physically random the way a coin flip is, but a well-implemented PRNG produces outputs statistically indistinguishable from true randomness for the purposes of a game economy. The pull mechanic typically works by drawing a value between 0 and 1 from the PRNG and comparing it against the current threshold: if the drawn value is below the pull’s probability, it’s a hit.
The distinction that actually matters isn’t between “fake” and “real” randomness, it’s between the randomness source and the probability function applied to it. A cryptographically sound PRNG can still produce unfair outcomes if the game applies the wrong threshold, and conversely a simple PRNG can produce fair outcomes if it’s correctly seeded and the threshold logic matches the published rate. That’s why the calculation work in this article matters more than debating the randomness source: the number that actually determines your odds is the pity-adjusted probability the developer chose to code in, not the entropy quality of the generator producing the roll.
Myths vs. Facts About Gacha Drop Rate Math
| Myth | Fact |
|---|---|
| A 0.6% rate means you’ll wait about 167 pulls on average. | Only true if there’s no pity system. With Genshin’s pity curve, the real expected value for any 5-star is roughly 62 pulls, not 167. |
| Losing a 50/50 means you effectively pay for two full unlucky runs. | The second run is guaranteed to hit the featured character, and the pity counter resets, so your second cycle averages the same ~62 pulls as your first, not a repeat of the worst case. |
| “Provably fair” is a marketing phrase every gacha game can claim. | It’s a specific technical property, typically involving published seeds or verifiable hash chains. Most mobile gacha games are disclosed, meaning the rates are published, but not provably fair in the cryptographic sense, since the server-side RNG can’t be independently verified pull by pull. |
| Every country requires gacha odds to be published by law. | China and South Korea have statutory disclosure requirements. Japan relies on industry self-regulation. The United States has no federal disclosure law at all. |
| If a game discloses a base rate, that’s the whole picture. | The base rate is only the floor. Soft-pity thresholds, hard-pity caps, and 50/50 or guarantee mechanics all change the real expected cost, and not every publisher discloses all of them. |
Frequently Asked Questions
Is a 0.6% drop rate the same as a 1-in-166 chance?
Only for a single pull, and only in a system with no pity. As a rough approximation, 1 divided by 0.006 equals about 167, which represents the average number of pulls needed in a flat-rate system with no acceleration. Once a pity curve is added, the real average drops well below that number, since the accelerating rate compresses the distribution toward the hard-pity pull count.
Why does the average pull count feel lower than 1 divided by the base rate?
Because soft pity does most of the statistical work. In Genshin’s case, a player who reaches pull 74 without a hit sees their odds jump from 0.6% to roughly 7%, and the rate keeps climbing every pull after that. That narrow high-probability window pulls the overall average down from a theoretical 167 pulls to roughly 62, even though most individual pulls in the sequence are still happening at the low base rate.
Do all gacha games use soft pity, or can a game run a flat rate with no curve?
Both models exist. NIKKE runs a flat 4% SSR rate with no acceleration at all, backstopped by a 200-pull deterministic mileage exchange instead of a rising probability curve. Genshin and Wuthering Waves both use accelerating soft-pity curves that ramp toward a hard guarantee. Neither approach is objectively more fair, they’re just different mathematical models for capping the downside of bad luck.
Can a developer secretly change drop rates after a game launches?
Technically, yes, since the probability table typically lives in server-side configuration that can be updated between patches. In practice, large-scale player tracking makes an undisclosed change statistically detectable over time, and in jurisdictions with legal disclosure requirements, an undisclosed change could also violate the law. That combination of community statistical monitoring and regulatory exposure in markets like South Korea is the main practical check against silent rate changes today.
Which countries legally require gacha rate disclosure in 2026?
China has required disclosure since a Ministry of Culture rule took effect in May 2017. South Korea made disclosure a statutory duty under an amended Game Industry Promotion Act effective March 22, 2024, with additional enforcement provisions added in 2025. Japan and the United States rely on industry self-regulation or general consumer-protection law rather than a dedicated gacha-disclosure statute.
Does losing a 50/50 always double the cost of getting a character?
No. Because the pity counter resets and the second cycle is guaranteed to land the featured character, the expected cost after losing a 50/50 is roughly the same as your first cycle’s average, not a full repeat of the worst-case 90-pull scenario. In Genshin’s math, that puts the overall expected cost at roughly 93 pulls rather than the 180-pull absolute ceiling some players assume.
Is a “provably fair” system the same thing as a published pity table?
No, and conflating the two is one of the most common mistakes players make. A published pity table tells you the intended probabilities. A provably fair system, the term used more precisely in crypto-native and casino-style games, lets a player independently verify that a specific past result matches a cryptographic seed the operator committed to in advance. Most mainstream gacha games disclose their rates but don’t offer that independent, pull-by-pull cryptographic verification, which means players are trusting the publisher’s disclosure rather than mathematically proving each individual roll.
