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Gacha probability explained
The actual math behind gacha rates. What a 0.6% rate really means, why pity changes expected costs, how consolidated rates are computed, and why set-completion mechanics explode. With an interactive calculator.
Contents6 sections
Gacha odds are one of the few parts of these games that can be reasoned about exactly. Publishers disclose base rates in most markets, the pull-by-pull rules are published or community-measured, and everything else is arithmetic. This page walks through that arithmetic. All worked examples use the published rates of real games, cited in the references.
What a base rate means
A 0.6% rate, the Genshin Impact character banner base rate, means each pull is an independent trial with a 0.6% chance of a 5-star. [1] Independence cuts both ways: previous failures do not make the next pull more likely (before soft pity intervenes), and a lucky early pull does not use up anything.
The chance of at least one success in n pulls is:
P(at least one) = 1 - (1 - p)^n
At p = 0.6% with no pity, the numbers are sobering:
| Pulls | Chance of at least one 5-star |
|---|---|
| 10 | 5.8% |
| 50 | 26.0% |
| 90 | 41.9% |
| 116 | 50.2% |
| 383 | 90.0% |
| 766 | 99.0% |
Half of all players would need 116 or more pulls for their first 5-star, and 1 in 100 would still have nothing after 766. This long tail is the problem pity systems exist to solve.
What pity does to the numbers
Hard pity truncates the distribution: in Genshin Impact a 5-star is guaranteed by pull 90, so the table above stops being true at 90, where the real probability becomes 100%. Soft pity (community-documented from roughly pull 74 in HoYoverse games) pulls most of the remaining tail forward, so in practice almost all 5-stars land between pull 70 and 85 for players who reach deep counts.
The published "consolidated rate" compresses all of this into one number. Genshin Impact discloses 1.6% consolidated for a 0.6% base, [1] and the reciprocal of the consolidated rate is the average cost:
average pulls per 5-star = 1 / 0.016 = 62.5
That average, 62.5 pulls, is the single most useful planning number for the HoYoverse-style games, and it comes straight from the official disclosure. Community statistics agree with it closely, which is a good sign the disclosed figure is honest.
Escalation systems do the same job with different shapes. Arknights publishes its rule outright: 2% base, rising 2 percentage points per pull after 50 dry pulls, reaching 100% at pull 99. [2] The resulting average is about 34.6 pulls per 6-star, computable directly from the published rule.
Guarantees compose into worst cases
Bounded systems allow exact worst-case budgeting. In a 90-pity, 50/50 game, the featured character costs at most 180 pulls: 90 to force a 5-star, lose the flip, 90 more for the guaranteed one. A spark is even simpler: 200 or 300 pulls buys the target outright, whatever the dice did. The comparison table lists worst cases per game with sources.
A useful habit: evaluate any banner by three numbers, the average cost (from the consolidated rate), the worst case (from pity and guarantees), and your pull budget. The calculator below computes the first two views for any rate.
Why set completion was banned
Kompu gacha demanded a complete set of random items. Completing a set is governed by collector mathematics, and the expected cost grows faster than intuition suggests: with N equally likely items, the expected number of relevant draws to complete the set is N times the harmonic number H(N). For a 10-item set that is about 29.3 relevant draws, and if each set item only appears once per 100 draws overall, the expectation is roughly 2,900 draws to complete by luck alone. The last missing item dominates the cost. Regulators did not need the formula, but the formula is why the player-harm stories kept happening, and Japan banned the mechanic in 2012.
Honest uncertainty
Two caveats apply to everything above. Soft pity thresholds in HoYoverse-style games are community-measured, not disclosed, so numbers like "ramp from pull 74" carry measurement uncertainty. And published rates are trusted inputs: outside China and South Korea, where audits back disclosure law, verification from outside is statistical. When this wiki cannot trace a number to a disclosure or a large-sample community measurement, it does not print the number.
Pull probability calculator
Model: independent pulls at a flat base rate, truncated by hard pity (the probability becomes 100% at the pity count). Real systems with soft pity give better odds than this near the pity counter, so treat these numbers as a floor. The math is computed in your browser; formulas are explained below.
References
- Genshin Impact in-game Wish Details rate disclosure HoYoverse. Accessed 2026-07-09. Source of the 0.6% base and 1.6% consolidated example rates.
- Arknights in-game Headhunting rules text Yostar / Hypergryph. Accessed 2026-07-09. Source of the published escalation rules used in the escalation section.
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