Calculadora de Distribuição Binomial Negativa
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
P(X=k) = C(k−1, r−1) · pʳ · (1−p)^(k−r)
About this calculator
This calculator determines the probability that the k-th independent Bernoulli trial results in the r-th success, following the negative binomial distribution. The formula used is P(X = k) = C(k−1, r−1) · pʳ · (1−p)^(k−r), where p is the probability of success per trial, r is the number of desired successes, and k is the total number of trials. The result represents the probability that exactly k trials are needed to achieve r successes.
How to use: enter the number of successes desired (r), the total number of trials (k), and the probability of success per trial (p) in the appropriate fields. The calculator will return the probability P(X = k). Ensure that r ≥ 1, k ≥ r, and 0 < p < 1. The result is useful for modeling situations where a fixed number of successes is awaited, such as in quality control, reliability analysis, or queueing studies.
Cautions: the negative binomial distribution assumes independent trials and constant probability of success. Do not confuse it with the binomial distribution (which fixes the number of trials). Verify that the data fits the 'waiting time' scenario until the r-th success. For p close to 0 or 1, results may be very small; consider using scientific notation.
Use cases: calculate the probability that the 3rd defective item appears on the 10th test, or the chance that the 5th customer buys after 12 approaches. It is also applicable in games of chance, such as the probability of getting the 2nd ace on the 7th card drawn from a deck with replacement.
Frequently asked questions
What is the difference between binomial and negative binomial distribution?
In the binomial, we fix the number of trials and count the number of successes. In the negative binomial, we fix the number of successes and count the number of trials needed to achieve them.
What does it mean if the probability is very low?
It indicates that it is unlikely that the r-th success occurs exactly at the k-th trial. This may be due to a small p or an unfavorable combination of r and k.
Can I use this calculator for p = 0 or p = 1?
No. p must be between 0 and 1 (exclusive). If p = 0, there are never successes; if p = 1, the first trial is already a success, so waiting for r successes makes no sense.
What if k is less than r?
The probability is zero, since it is impossible to obtain r successes in fewer than r trials. The calculator will return an error or zero, depending on validation.
Is this distribution the same as Pascal distribution?
Yes, the negative binomial distribution with integer r is known as the Pascal distribution.