Negative Binomial Distribution Calculator

Formula

How this is calculated

The negative binomial (Pascal) distribution models the number of failures X = k that occur before the r-th success, where each independent trial succeeds with probability p. The probability mass function is P(X = k) = C(k + r − 1, k) · p^r · (1 − p)^k for k = 0, 1, 2, …. The binomial coefficient counts the arrangements of the first k failures and r − 1 successes among the first k + r − 1 trials; the final trial is always the r-th success. This calculator evaluates the coefficient through the log-gamma function so it stays numerically stable for large r and k.

Enter r (successes needed, a positive integer), p (per-trial success probability between 0 and 1), and k (failures). The mode selector chooses between an exact point probability P(X = k), the lower cumulative P(X ≤ k), and the upper tail P(X ≥ k) = 1 − P(X ≤ k − 1). The mean number of failures is r(1 − p)/p and the variance is r(1 − p)/p², so the spread grows quickly as p approaches 0.

Assumptions: trials are independent and identically distributed, p is constant, and counts are non-negative integers (k is rounded). The result is undefined when p ≤ 0 or p > 1; as p → 1 the distribution collapses toward k = 0 because successes arrive almost immediately.