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.
P(X = x)
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P(X ≤ x)
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P(X ≥ x)
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P(x₁ ≤ X ≤ x₂)
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Mean (μ)
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Variance (σ²)
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Standard Deviation (σ)
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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.
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.
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.
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.
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.
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.
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.
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.