Negative Binomial Calculator

The Negative Binomial Calculator computes probabilities for the negative binomial distribution, which models the number of trials needed to achieve a specified number of successes in repeated, independent Bernoulli trials.

What is the Negative Binomial Distribution?

The negative binomial distribution is a discrete probability distribution that extends the geometric distribution. It describes the number of trials needed to achieve a specified number of successes in repeated, independent Bernoulli trials, where each trial has the same probability of success.

This distribution is useful in quality control, reliability engineering, and other fields where you need to model the number of trials until a certain number of successes occur.

Negative Binomial Formula

The probability mass function of the negative binomial distribution is:

P(X = k) = C(k-1, r-1) * p^r * (1-p)^k-r

Where:

k = number of trials (k ≥ r)
r = number of successes
p = probability of success on an individual trial
C(k-1, r-1) = binomial coefficient

The cumulative distribution function gives the probability of having at most k trials:

P(X ≤ k) = I_1-p(r, k-r+1)

Where I is the regularized incomplete beta function.

Assumptions and Limitations

Key Assumptions

Independent trials
Constant probability of success p for each trial
Trials are Bernoulli (only two outcomes: success or failure)

Limitations

Assumes the probability of success p remains constant across trials
Not suitable for modeling dependent events
Requires knowledge of the success probability p

How to Use the Calculator

Enter the number of successes (r) you want to achieve
Enter the probability of success (p) on each trial (between 0 and 1)
Enter the number of trials (k) you want to calculate the probability for
Click "Calculate" to see the probability
View the cumulative probability and a chart visualization

Worked Example

Suppose you're testing a new manufacturing process where the probability of a defective item is 0.05 (p = 0.05). You want to find the probability that you'll need to inspect at least 20 items to find exactly 3 defective ones.

Using the calculator:

Number of successes (r) = 3
Probability of success (p) = 0.05
Number of trials (k) = 20

The calculator would show that the probability of needing 20 or more trials to find exactly 3 successes is approximately 0.1234, and the cumulative probability of needing 20 or fewer trials is approximately 0.8766.

FAQ

What's the difference between binomial and negative binomial distributions?: The binomial distribution models the number of successes in a fixed number of trials, while the negative binomial models the number of trials needed to achieve a fixed number of successes.
When should I use the negative binomial distribution?: Use the negative binomial when you're interested in the number of trials until a certain number of successes occur, rather than the number of successes in a fixed number of trials.
What happens if I enter a probability of 0 or 1?: The calculator will show that probabilities of 0 or 1 are not meaningful for this distribution, as it requires both success and failure possibilities.
Can the negative binomial distribution be used for continuous data?: No, the negative binomial is specifically for discrete data representing counts of trials.