Geometric Distribution Calculator

About this calculator

The geometric distribution models the number of independent trials needed to get the first success, when each trial has the same success probability p. The probability that success first happens exactly on trial k is P(X=k) = (1−p)^(k−1) × p, the probability it happens within the first k trials is 1 − (1−p)^k, and the expected (average) number of trials needed is 1/p.

It’s the standard model whenever you’re asking “how long until the first hit” under repeated, independent chances: quality-control engineers use it to estimate how many items get inspected before the first defect turns up, sales teams use it to model how many cold calls precede the first conversion, and reliability engineers use it for the number of test cycles before a first component failure. In sports analytics it explains things like the expected number of free throws before the first miss for a player with a known shooting percentage.

This calculator takes your success probability p and trial count k and returns all three quantities — the exact-trial probability, the cumulative probability, and the expected number of trials — in one step.