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Generated on: July 20, 2026 at 09:33 PM

Negative Binomial Distribution Calculator

Inputs

Number of successes3
Success probability0.4
Number of failures4

Visualization

PMF value 0

Generated with OneCalc · https://onecalc.app/en/math/negative-binomial-distribution

Mathematics

Negative Binomial Distribution Calculator

Calculate the probability of exactly k failures before the r-th success, plus the cumulative probability, mean, and variance of a negative binomial distribution.

Inputs

Parameters

≥ 1
0 – 1

Results

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Details

Probabilities

\begin{aligned} P(X=k) &= \binom{k+r-1}{k} \, p^r (1-p)^k \\ &= \binom{ 4+3-1 }{ 4 } \, 0.4^{ 3 } (1-0.4)^{ 4 } \\ &= ? \end{aligned}
\begin{aligned} P(X \leq k) &= \sum_{i=0}^{k} \binom{i+r-1}{i} \, p^r (1-p)^i \\ &= ? \end{aligned}
\begin{aligned} \mu &= \dfrac{r(1-p)}{p} \\ &= \dfrac{ 3(1-0.4) }{ 0.4 } \\ &= ? \end{aligned}
\begin{aligned} \sigma^2 &= \dfrac{r(1-p)}{p^2} \\ &= \dfrac{ 3(1-0.4) }{ 0.4^2 } \\ &= ? \end{aligned}

Distribution

PMF value 0

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Negative Binomial Distribution

The negative binomial distribution is a discrete probability distribution that models the number of failures that occur before a fixed number of successes is reached in a sequence of independent Bernoulli trials. Each trial shares the same success probability p and is independent of all others. It appears in reliability testing, sales and marketing analysis, epidemiology, and any process that continues until a target number of successes has accumulated.

Formula

This calculator uses the failures-before-success convention: X is the number of failures k observed before the r-th success, so X ≥ 0. The probability mass function (PMF) and cumulative distribution function (CDF) are:

P(X=k)=(k+r−1k) pr(1−p)k,k=0,1,2,…P(X = k) = \binom{k + r - 1}{k} \, p^{r} (1-p)^{k}, \quad k = 0, 1, 2, \ldots P(X≤k)=∑i=0k(i+r−1i) pr(1−p)iP(X \leq k) = \sum_{i=0}^{k} \binom{i + r - 1}{i} \, p^{r} (1-p)^{i}

The mean and variance of X under this convention are:

μ=r(1−p)p,σ2=r(1−p)p2\mu = \frac{r(1-p)}{p}, \qquad \sigma^2 = \frac{r(1-p)}{p^2}

A second convention defines X as the total number of trials needed to obtain r successes (X ≥ r). That variable equals k + r, so its PMF is the same shape shifted by r, and its mean is r/p. The two conventions describe the same experiment.

Worked example

A quality inspector accepts a batch once three units have passed inspection. Each unit passes independently with probability p = 0.40. What is the probability that exactly four units fail before the third unit passes?

Using r = 3, p = 0.40, and k = 4:

P(X=4)=(4+3−14)×0.403×0.604=15×0.064×0.1296=0.1244P(X = 4) = \binom{4 + 3 - 1}{4} \times 0.40^{3} \times 0.60^{4} = 15 \times 0.064 \times 0.1296 = 0.1244

There is a 12.44 % chance that exactly four units fail before the third pass. The cumulative probability of four or fewer failures is:

P(X≤4)=∑i=04(i+2i)×0.403×0.60i=0.5801P(X \leq 4) = \sum_{i=0}^{4} \binom{i+2}{i} \times 0.40^{3} \times 0.60^{i} = 0.5801

So there is a 58.01 % chance that no more than four units fail. The expected number of failures is μ = 3 × 0.60 / 0.40 = 4.5, with variance σ² = 3 × 0.60 / 0.40² = 11.25 and standard deviation σ ≈ 3.35.

Formula derivation

For exactly k failures to precede the r-th success:

  1. The final trial must be the r-th success, which occurs with probability p.
  2. Among the preceding k + r − 1 trials, exactly k are failures and r − 1 are successes. The number of ways to arrange them is C(k+r−1, k).
  3. Those k failures each occur with probability 1−p and the r − 1 successes each with probability p. Combined with the final success, the trial outcomes contribute pʳ · (1−p)ᵏ.

Multiplying the ordering count by the outcome probability gives C(k+r−1, k) · pʳ · (1−p)ᵏ. The CDF is the running sum of the PMF from 0 up to k.

Relationship to other distributions

The negative binomial distribution generalises the geometric distribution: setting r = 1 counts the failures before the first success, and P(X = k) = (1−p)ᵏ · p. Equivalently, a negative binomial random variable is the sum of r independent geometric random variables that share the same success probability p.

It is the complement of the binomial distribution. The binomial fixes the number of trials and treats successes as random; the negative binomial fixes the number of successes and treats the number of failures (or trials) as random. When the number of successes r is large and p is close to 1, the distribution becomes approximately normal. When r is not required to be an integer, the same formula with the gamma function defines the closely related Polya distribution, widely used to model overdispersed count data.

Practical applications

  • Quality control: modelling how many defective units are inspected before a fixed number of acceptable units is found.
  • Sales and marketing: estimating the number of unsuccessful contacts before a set number of conversions is achieved.
  • Epidemiology and ecology: fitting overdispersed count data, where the variance exceeds the mean, better than a Poisson model.
  • Reliability engineering: computing the expected number of failed components before a redundant system accumulates enough working parts.

Frequently Asked Questions (FAQ)

What is the negative binomial distribution?

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 Bernoulli trials. Each trial has the same success probability p. For example, with r = 3 and p = 0.4, the distribution describes how many failed attempts occur before the third success.

Unlike the binomial distribution, which fixes the number of trials and counts successes, the negative binomial fixes the number of successes and counts the trials (or failures) needed to reach it.

Which convention does this calculator use?

This calculator counts the number of failures k before the r-th success, so k ≥ 0 and P(X = k) = C(k+r−1, k) · pʳ · (1−p)ᵏ. The mean is r(1−p)/p and the variance is r(1−p)/p². A common alternative defines X as the total number of trials needed to obtain r successes; that variable equals k + r, shifting the support to X ≥ r and the mean to r/p. The two conventions describe the same experiment and differ only by the constant r.

How does it differ from the binomial distribution?

The binomial distribution fixes the number of trials n and treats the number of successes as random. The negative binomial does the opposite: it fixes the number of successes r and treats the number of failures (or trials) as random. In the binomial the stopping rule is "stop after n trials"; in the negative binomial it is "stop after r successes". They share the same per-trial success probability p and the same independence assumptions.

How does it relate to the geometric distribution?

The geometric distribution is the special case of the negative binomial with r = 1: the number of failures before the first success. Setting r = 1 in P(X = k) = C(k+r−1, k) · pʳ · (1−p)ᵏ gives P(X = k) = (1−p)ᵏ · p, the geometric PMF. Equivalently, a negative binomial random variable is the sum of r independent geometric random variables with the same p.

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