Why the Average Misleads You
The most commonly cited number in gacha discussion is the average pulls to success, 1/p. But the geometric distribution has a long, heavy tail, so budgeting against the average ignores the very real risk of falling into the unlucky tail. With a 1% drop rate the average is 100 pulls, yet 1 in 4 players need over 138 and 1 in 10 must endure more than 229 before their first hit.
Reading the Percentile Table
"Top 10% unlucky" means that if 100 players each ran the same banner, the 10 unluckiest would not have succeeded by that pull count. A 90% safety budget is roughly 2.3x the average. Games with pity systems effectively cap this tail - your worst case becomes the smaller of the 99% percentile and the pity number.
Using This Before You Spend
Before chasing a limited unit, compute the budget you would need to survive a 90% or 99% confidence run. Many players assume the average will suffice and end up spending 2-3x more when their RNG lands in the bad tail. Decide your stop-loss in advance and stick to it.
FAQ
No. The math assumes a fixed probability. Soft pity or rate-up systems need a separate simulation.
The median is roughly 0.69/p, about 69% of the average. Half of all players actually succeed below the mean.
The probability that anyone in a group of n hits the target goes to 100% much faster than for any single player.