At Least Once Probability Calculator

About this calculator

When an event has probability p of happening on a single independent trial, the probability that it happens at least once across n trials is easiest to find through the complement: instead of adding up every way it could succeed once, twice, or more, you calculate the probability it never happens — (1−p)ⁿ — and subtract that from 1. So P(at least one) = 1 − (1−p)ⁿ.

This formula underlies risk assessments across many fields: reliability engineers use it to estimate the chance that at least one component fails over n operating cycles, epidemiologists apply it to find the odds that at least one person in a group of n tests positive, and security teams use it to model the probability that at least one intrusion attempt out of many succeeds against a system with a known per-attempt breach rate. It is also the reasoning behind gambling odds, like the chance of rolling at least one six in several dice rolls.

Enter your single-trial probability and number of trials, and this calculator applies the complement rule directly to give you the combined probability.