What does the negative binomial distribution model?

It models the number of failures that occur before a fixed number of successes (r) is achieved, across a sequence of independent trials each with the same success probability p.

What is the negative binomial probability formula?

P(X = k) = C(k+r−1, k) × (1−p)ᵏ × pʳ, where k is the number of failures, r is the number of successes needed, and p is the probability of success on each trial.

How is this different from the binomial distribution?

The binomial distribution fixes the number of trials and asks about the number of successes. The negative binomial distribution fixes the number of successes needed and asks about the number of failures along the way — the number of trials itself is random.

How is this different from the geometric distribution?

The geometric distribution is a special case of the negative binomial where r = 1 — it models the number of failures before the very first success. The negative binomial generalizes this to any number of required successes.

What are the mean and variance of a negative binomial distribution?

Mean = r(1−p)/p, and variance = r(1−p)/p². Both grow as the required number of successes (r) increases or the success probability (p) decreases.

Why must r be a positive whole number?

r represents a count of successes you're waiting to reach, so it must be a positive integer — you can't require 'half a success' in this model.

Can p equal exactly 1?

Yes — if p = 1, every trial succeeds, so zero failures occur before r successes, meaning P(X = 0) = 1 and P(X = k) = 0 for any k > 0.

Where is the negative binomial distribution used in real life?

It's used to model sales calls needed before closing r deals, defective items encountered before finding r good ones, and overdispersed count data in insurance and epidemiology where the simple Poisson model underestimates variability.

Negative Binomial Distribution Calculator — Find P(X = k) Failures Before r Successes

Our Negative Binomial Distribution Calculator finds the probability of observing exactly k failures before achieving the rth success, in a sequence of independent trials each with success probability p, using the formula P(X = k) = C(k+r−1, k)(1−p)ᵏpʳ. It also reports the distribution's mean (r(1−p)/p) and variance (r(1−p)/p²) — useful for modeling 'how many failures until we succeed r times' scenarios like sales calls before r deals close, or defective parts before r good ones are found.

Quick Answer

The negative binomial distribution gives the probability of k failures before the rth success: P(X = k) = C(k+r−1,k)(1−p)ᵏpʳ. Enter r, p, and k below to get the exact probability along with the distribution's mean and variance.

Successes needed (r)

Probability of success (p)

Failures (k)

Enter r, p, and k, then click Calculate.

How to Use the Negative Binomial Distribution Calculator — Failures Before Success

1
Enter the number of successes needed (r).
2
Enter the probability of success on a single trial (p), between 0 and 1.
3
Enter the exact number of failures (k) you want the probability for.
4
Click 'Calculate' to get P(X = k) along with the distribution's mean and variance.

Why Use Negative Binomial Distribution Calculator — Failures Before Success?

The negative binomial distribution flips the usual binomial question around: instead of asking how many successes occur in a fixed number of trials, it asks how many failures happen before a fixed number of successes is reached — the natural model for 'keep trying until you succeed r times.' The formula involves a combination term with a shifted index, C(k+r−1, k), which is easy to set up incorrectly by hand. This calculator computes the exact probability and the distribution's summary statistics instantly, with the formula shown step by step.

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