The Math Behind Gacha Pulls
For a pull with drop rate p, the average number of attempts to score your first success is 1/p. A 1% drop averages 100 pulls; a 0.6% drop averages about 167 pulls. Multiply by the price per pull to estimate your expected spend - but remember this is the average, not a guarantee, and real outcomes vary heavily.
Why the Average Is a Trap
Gacha follows the geometric distribution, which has a long tail. At a 1% drop, the average is 100 pulls, but only 50% of players hit by pull 69, while 25% need over 138 and 10% need over 229 pulls to win. Budgeting against the average alone can blow up if your luck lands in the unlucky tail. To plan with 90% confidence, expect to spend roughly 2.3x the average.
Pity and Hard Caps
Most modern games include pity systems that guarantee a drop after a fixed number of fails. When pity is shorter than the 99% percentile, pity becomes the real maximum. If pity is 100 pulls, no run can exceed 100 attempts, regardless of probability.
FAQ
Pre-apply the discount to the single-pull price (for example, divide a 10-pull bundle price by 10) and use that number.
No. The numbers are pure probability. If a pity cap exists, use the lower of the 99% percentile and the pity number as your worst case.
It works fine. A 0.1% drop averages 1,000 pulls and needs about 2,300 for 90% confidence.