Negative Binomial Distribution Calculator

Work out negative binomial distribution instantly with clear inputs, formula shown and shareable results.

About Negative Binomial Distribution Calculator

The negative binomial gives the probability that the rth success falls exactly on trial n: C(n−1, r−1)pʳ(1−p)^(n−r). Its mean r/p reflects that each success takes 1/p trials on average.

Formulas

Negative binomial pmf

P(X = n) = C(n−1, r−1)·pʳ·(1−p)^(n−r)

Mean and variance

E = r/p, Var = r(1−p)/p²

Worked example

With the default values below, Negative Binomial Distribution Calculator returns P(the rth success occurs on trial n) of 0.0800483796. Change any input above and every figure here is recalculated in your browser.

Inputs used

Required successes raccepts 1–200: 3
Success probability paccepts 0.01–100 %: 30 %
Total trials naccepts 1–2,000: 10

Results

P(the rth success occurs on trial n): 0.0800483796
P(r successes within n trials): 0.6172172136
Expected trials to reach r successes: 10.000000
Variance of the trial count: 23.333333

Step by step

1Values used
2Negative binomial pmf
3Mean and variance
4P(the rth success occurs on trial n)= 0.0800483796
5P(r successes within n trials)= 0.6172172136
6Expected trials to reach r successes= 10.000000
7Variance of the trial count= 23.333333

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For r = 3, p = 0.3 and n = 10?

P ≈ 0.0800, and 3 successes are expected to take about 10 trials.

How does it differ from the binomial?

The binomial fixes the trial count and counts successes; the negative binomial fixes the success count and counts trials.