Negative Binomial Distribution

Calculate the probability of failures before a target number of successes.

Parameters

Target Successes (r)

Success Probability (p)

Failures to Test (k)

Distribution Results

Probability P(X=k)13.67%

Expected Failures5.00

Std Deviation3.16

Variance10.00

Ratio (Var/Mean)2.00

Overview

The Negative Binomial distribution answers: "What is the probability that I will fail exactly k times before I reach my r-th success?" It's a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified number of successes occurs.

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Pro Tips

Use this when you have a 'stopping rule' based on a fixed number of successes.
Ensure the success probability (p) is between 0 and 1.
The value 'r' must be a positive integer.

Fun Facts

"In ecology, the Negative Binomial is used to model the distribution of species where individuals tend to clump together."
"It is often used as an alternative to the Poisson distribution when the variance is greater than the mean (overdispersion)."
"The Geometric distribution is a special case of the Negative Binomial where r = 1."

Formula: P(X=k) = (k+r-1)C(k) * pʳ * (1-p)ᵏ

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