Generic gacha math · user-supplied inputs

Pulls Needed for a Target Chance

Invert the usual probability formula to solve for pull count instead of testing budgets one by one.

Quick answer

Example: At a fixed 1% rate, the minimum pull count for at least a 90% chance is 230. The result rounds upward and reports the probability at that integer count.

Minimum pulls —

Mean pulls to first success —

Chance at rounded pull count —

What this calculator does

This calculator works backward from a desired probability. It solves the no-success equation with logarithms, then rounds up because a pull count must be a whole number. The achieved probability is shown after that rounding step.

The mean waiting time is also displayed, but it answers a different question. A mean of 100 pulls at a 1% rate does not mean 100 pulls provide certainty; independent fixed-rate attempts retain a nonzero miss chance.

Targets are limited below 100% because a fixed probability smaller than one never reaches mathematical certainty in a finite number of independent trials.

Formula

pulls = ceil(log(1 − target chance) ÷ log(1 − drop rate))

Example: At a fixed 1% rate, the minimum pull count for at least a 90% chance is 230. The result rounds upward and reports the probability at that integer count.

FAQ

Why can I not request exactly 100%?

With a constant rate below 100%, every finite pull count retains some miss probability.

Why is the mean lower than a high-confidence pull count?

The geometric waiting-time distribution has a long tail; high confidence must cover more of that tail than the mean does.

Does this account for pity?

No. It is the inverse formula for a fixed independent rate.

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