Negative Binomial Distribution Calculator

About this calculator

The negative binomial distribution models the number of failures that occur before a fixed number of successes, r, is reached in a sequence of independent trials each with success probability p — for example, how many times you would expect to miss a free throw before making your 5th successful one. Its mean is r(1−p)/p and its variance is r(1−p)/p², and setting r = 1 reduces it to the simpler geometric distribution, which just counts failures before the first success.

It differs from the ordinary binomial distribution in what is held fixed: binomial fixes the number of trials and counts successes, while negative binomial fixes the number of successes and counts trials (or failures) needed to get there. Beyond coin-flip style examples, the distribution is widely used in reliability engineering to model the number of failures before a set number of successful units, in sports analytics for streaks, and in ecology and epidemiology to model overdispersed count data — counts with more variance than a plain Poisson distribution would predict, such as disease case counts across regions.

Enter the number of required successes r, the success probability p, and the number of failures k, and the calculator returns the probability along with the distribution's mean and variance.