Negative Binomial Distribution Calculator

Negative Binomial Distribution Calculator. Enter the number of successes (r), probability of success (p), and number of trials (x) to compute negative binomial probabilities. Your results include P(X = x), P(X ≤ x), P(X ≥ x), plus the mean and standard deviation of the distribution — all based on the Pascal distribution formula. Also try the Multinomial Distribution Calculator.

Results

P(X = x)

P(X ≤ x)

P(X ≥ x)

P(x₁ ≤ X ≤ x₂)

Mean (μ)

Variance (σ²)

Standard Deviation (σ)

Results Table

What is a negative binomial experiment?

A negative binomial experiment is a statistical experiment with a fixed probability of success p on each independent trial, repeated until exactly r successes occur. The outcome of interest is X — the total number of trials needed to achieve r successes. Each trial must be independent, and the probability of success must remain constant. See also our Hypergeometric Distribution Calculator.

What is the negative binomial distribution?

The negative binomial distribution (also called the Pascal distribution) describes the probability of achieving the r-th success on exactly the x-th trial. Its PMF is P(X=x) = C(x-1, r-1) × p^r × (1-p)^(x-r), where x ≥ r, r ≥ 1, and 0 < p < 1. It generalizes the geometric distribution to multiple required successes.

What is the number of trials (x) in the negative binomial distribution?

The number of trials x represents the total number of independent trials performed until the r-th success is observed. Because we need at least r successes, x must always be greater than or equal to r. For example, if you need 3 heads in coin flips, x is the flip number on which you get your 3rd head.

What is the probability of success on a trial (p)?

The probability of success p is the fixed likelihood of success on any single trial, where 0 < p < 1. It must remain constant across all trials for the negative binomial distribution to apply. For instance, a fair coin has p = 0.5, and a biased die showing a six has p ≈ 0.167. You might also find our calculate Discrete Uniform Distribution P(X = x) useful.

How is the negative binomial distribution related to the geometric distribution?

The geometric distribution is a special case of the negative binomial distribution where r = 1 — meaning you are waiting for just the first success. When r = 1, the negative binomial PMF reduces exactly to the geometric PMF: P(X = x) = (1-p)^(x-1) × p.

How is the negative binomial distribution different from the binomial distribution?

In a binomial experiment, the number of trials is fixed and you count the number of successes. In a negative binomial experiment, the number of successes is fixed (r) and you count the number of trials needed to reach that target. They are complementary perspectives on the same type of repeated Bernoulli trials.

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

For a negative binomial distribution with parameters r and p (where X = number of trials), the mean is μ = r/p and the variance is σ² = r(1-p)/p². The standard deviation is σ = √(r(1-p)/p²). These formulas assume X counts total trials; if X counts only failures, the mean is r(1-p)/p.

What does P(X ≤ x) represent in the negative binomial calculator?

P(X ≤ x) is the cumulative distribution function (CDF) — the probability that the r-th success occurs on or before the x-th trial. It is computed by summing P(X = k) for all k from r up to x. This is useful when you want to know the likelihood of achieving your goal within a certain number of attempts.