Negative Binomial Distribution Calculator

The negative binomial distribution, in its "trial of the r-th success" form, models repeating independent trials with a fixed per-trial success probability p — the same fixed-p assumption as this site's Binomial Probability calculator — until a target number of successes r is reached, rather than stopping after a fixed number of trials. The probability the r-th success lands exactly on trial n is P(N = n) = C(n − 1, r − 1) × p^r × (1 − p)^(n − r): trial n itself must be a success, and the other r − 1 successes can fall anywhere among the preceding n − 1 trials, which is what the binomial coefficient counts.

Setting the target successes r to 1 reduces this exactly to the geometric distribution — the trial number of the first success. The expected (mean) trial number for the r-th success to occur is r ÷ p, and its variance is r × (1 − p) ÷ p², both standard closed-form results. This is distinct from the site's Binomial Probability calculator, which fixes the number of trials and asks about the number of successes — negative binomial flips that around, fixing the number of successes and asking about the number of trials.