Expected Number of Trials Calculator

About this calculator

When each independent trial has a fixed success probability p, the number of trials needed until the first success follows the geometric distribution. Its expected value is E[X] = 1/p, its median is the smallest integer m satisfying (1−p)^m ≤ 0.5, and the number of trials needed to reach 90% confidence of at least one success is the smallest n satisfying (1−p)^n ≤ 0.10.

Statisticians and quality control engineers use these figures to plan sampling or inspection processes, game designers use them to tune random item drop rates so players get a fair sense of how many attempts a rare drop should take, and anyone reasoning about repeated independent attempts — from A/B test iterations to games of chance — uses the geometric distribution to separate an average outcome from a reasonably confident upper bound.