Calculate Negative Binomial Distribution

The negative binomial distribution is a probability distribution that models the number of trials needed to achieve a given number of successes in repeated, independent Bernoulli trials. This calculator helps you compute probabilities for the negative binomial distribution based on your parameters.

What is the Negative Binomial Distribution?

The negative binomial distribution describes the probability of having a certain number of failures before achieving a specified number of successes in a series of independent Bernoulli trials. It's commonly used in quality control, reliability engineering, and other fields where counting events until a certain number of successes occur is important.

Key Characteristics:

Models the number of trials until a specified number of successes
Parameters: number of successes (k), probability of success (p)
Discrete probability distribution
Right-skewed distribution

Unlike the binomial distribution, which models the number of successes in a fixed number of trials, the negative binomial distribution models the number of trials needed to achieve a fixed number of successes.

Formula and Calculation

The probability mass function for the negative binomial distribution is given by:

P(X = x) = C(x-1, k-1) × p^k × (1-p)^x-k

Where:

x = number of trials (x ≥ k)
k = number of successes
p = probability of success on an individual trial
C(x-1, k-1) = combination of (x-1) things taken (k-1) at a time

This formula calculates the probability of having exactly x trials, including exactly k successes, where the last trial is a success.

Assumptions:

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

How to Use the Calculator

To use the negative binomial distribution calculator:

Enter the number of successes (k) you want to achieve
Enter the probability of success (p) on each trial
Specify the number of trials (x) you want to calculate the probability for
Click "Calculate" to see the probability
View the result and chart visualization

The calculator will display the probability of exactly x trials occurring, including exactly k successes, with the last trial being a success. The chart shows the probability distribution for different values of x.

Examples and Applications

Here are some practical applications of the negative binomial distribution:

Application	Description
Quality Control	Modeling the number of defective items produced until a certain number of good items are found
Reliability Engineering	Predicting the number of system failures before a specified number of successful operations
Medical Trials	Calculating the number of patients needed to observe a certain number of successes (e.g., cures)
Sports Analytics	Modeling the number of games needed to achieve a certain number of wins

For example, in quality control, you might want to know the probability of finding 5 defective items before finding 10 good items, given that each item has a 90% chance of being good.

Frequently Asked Questions

What is 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 distribution 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 distribution 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 are the parameters of the negative binomial distribution?

The negative binomial distribution has two main parameters: the number of successes (k) and the probability of success on each trial (p).

How does the negative binomial distribution differ from the geometric distribution?

The geometric distribution is a special case of the negative binomial distribution where k=1 (only one success is needed).