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Gacha Probability Calculator

Enter your per-pull rate and how many pulls you plan to do. You get the chance of at least one success, the average number of pulls to your first hit, and the pull counts that actually give you 50%, 90% and 99% confidence.

The unmodified rate shown in the game's details screen, e.g. 0.6 for a 5-star.
Try 10, 90 or 180 to compare realistic budgets.
Independent pulls only. If your game has a pity system that raises the rate, use the pity calculator instead — it models the real curve.
Chance of at least one hit

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Chance of nothing—
Average pulls to first hit—
Median pulls to first hit—

What this calculator actually computes

Every pull is an independent event with the same chance p. The probability that you get nothing in n pulls is the chance of missing every single time:

P(no hit) = (1 − p)n

So the probability of at least one hit is simply the complement:

P(at least one) = 1 − (1 − p)n

This is a geometric distribution. The expected number of pulls to your first hit is 1 / p, and the median — the point where half of players have hit at least once — is log(0.5) / log(1 − p).

A worked example

At 0.6% per pull over 90 pulls:

1 − (1 − 0.006)90 = 1 − 0.99490 ≈ 41.8%

That number surprises people. Ninety pulls sounds like a lot, but it still leaves roughly a 58% chance of walking away with nothing. To reach 90% confidence at 0.6% you need 383 pulls; for 99% you need 766.

The mistake almost everyone makes

Players often read "0.6%" as "one in 167 pulls". That is the average, not a guarantee, and the distribution is heavily skewed: a lucky minority pulls early, which drags the mean down while most players sit above it. At 0.6%:

ConfidencePulls neededChance of nothing after that many pulls
50%11650%
75%23125%
90%38310%
99%7661%

Base rate vs. consolidated rate

This is the single most common source of bad math. Games often publish two numbers:

  • Base rate — the chance of any single pull, before pity. For Genshin and Honkai: Star Rail this is 0.6%.
  • Consolidated rate — the long-run average after pity is included. The same games list 1.6%.

If you plug the consolidated rate into a plain independent-pull formula you will understate how bad an unlucky streak can be, because the consolidated figure only exists because of the pity floor. Use the base rate here, and use the pity calculator when the game has a built-in floor.

Rule of thumb: with independent pulls, doubling the pulls never doubles your chance. Going from 90 to 180 pulls at 0.6% takes you from 41.8% to 66.2%, not 84%.

When this model is wrong

Three cases where the plain formula will mislead you:

  1. Soft pity. The rate climbs near the ceiling, so late pulls are worth far more than early ones. Using a flat rate massively understates your odds in the 70–90 pull range.
  2. Guaranteed featured. Losing a 50/50 sets up a guaranteed pull, which roughly halves your cost for the second copy of a character.
  3. Multi-pull bonuses. If a ten-pull includes a guaranteed 4-star, that changes the 4-star math (not the 5-star math).