Gacha maths

The unlucky tail: the number the average hides

Two games can average the same number of pulls and cost wildly different amounts when it goes badly. The gap between a typical run and an unlucky one is the number worth planning against, and here it is for every game.

By the GachaData team Updated 3 min read

Every gacha comparison you will read leads with an average. It is the wrong number to save against.

An average is what happens across thousands of accounts. What happens to yours is one draw from a distribution, and in a gacha that distribution is badly lopsided: a long right tail where the unlucky runs live. Plan against the average and you will be short roughly half the time.

The useful pair of numbers is the typical run and the unlucky one — the 50th and 90th percentile. One in ten players does worse than the unlucky column.

Typical, average, unlucky

GameTypical (50%)AverageUnlucky (90%)Guaranteed bySpread
CookieRun: Crumble116167383never267
Dragon Ball Z Dokkan Battle97140322never225
Umamusume: Pretty Derby93133306not carried213
Fate/Grand Order87116287330200
Azur Lane5876191200133
Hololive Dreams6987200200131
Limbus Company96106200200104
Blue Archive99108200200101
Arknights5769142never85
Genshin Impact809115618076
Silver And Blood3650111never75
Duet Night Abyss819415618075
Honkai: Star Rail819415618075
Zenless Zone Zero819415618075
Persona5: The Phantom X748814516071
Wuthering Waves758314516070
DIGIMON UP13912720020061
Reverse: 1999706512314053
Blue Protocol: Star Resonance686571803

All figures are pulls to the banner’s featured unit from zero, with no carried-over pity, and already account for each game’s ramp or spark. Spread is the unlucky figure minus the typical one — how much extra a bad run costs. Sorted widest spread first.

The spread is the story

A guarantee is a spread cap

Look at what the Guaranteed by column does to the Spread column. The games with no guarantee occupy almost the whole top of this table. Where a game guarantees the unit, the unlucky case cannot run past the ceiling, so the spread is bounded by construction. Where it does not, nothing bounds it — the 90th percentile is simply where the distribution happens to be, and the 99th is further still.

This is why a guarantee is worth more than a good average: it converts an open-ended risk into a known worst case you can actually save for.

Two games can tie on the average and not be alike

The averages compress games that behave very differently. Read the typical and unlucky columns together and pairs that look identical in a one-number ranking come apart immediately — a game whose ramp is tight lands nearly everyone in a narrow band, while a flat-rate game scatters the same average across a range several times wider.

What to actually budget

If you are saving for a specific unit, the unlucky column is the honest target and the guarantee column is the ceiling. The average is useful for comparing games, and for nothing else. Each game’s own Pull Planner runs these numbers for any starting pity count you already have.

What this does not say

These are pull counts, not money: a pull costs a different amount in every game, which is its own comparison. The percentiles assume you start from zero pity and stop the moment you get the unit. And they describe the banner type each game leads with — a game running several pools with different rules is represented by one of them, named on its own page.

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