Negative Binomial PMF Calculator

About this calculator

The negative binomial distribution models the number of failures k observed before achieving a fixed number of successes r, in a sequence of independent trials each with success probability p. Its probability mass function is P(X=k) = C(k+r−1, k) · p^r · (1−p)^k, with mean μ = r(1−p)/p and variance σ² = r(1−p)/p² — a distribution that, unlike the Poisson, allows variance to exceed the mean, making it useful for “overdispersed” count data.

This calculator takes the number of failures, the target number of successes, and the per-trial success probability, and returns the PMF along with the distribution's mean and standard deviation. Statisticians and actuaries use it to model overdispersed count data such as insurance claim frequencies, and quality-control engineers use it to estimate how many defective units to expect before hitting a target number of good ones in a production run.