Distribution Calculator Negative Binomial

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 and expected values for the Negative Binomial distribution.

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 the number of trials until a certain number of successes is important.

Key Characteristics

Models the number of trials until a specified number of successes
Defined by two parameters: number of successes (k) and probability of success (p)
Right-skewed distribution
Mean = k/p
Variance = k(1-p)/p²

When to Use

The Negative Binomial Distribution is appropriate when:

You need to model the number of trials until a certain number of successes
Trials are independent
Probability of success is constant across trials
You're interested in the number of failures before the kth success

How to Use This Calculator

To use the Negative Binomial Distribution Calculator:

Enter the number of successes (k) you're interested in
Enter the probability of success (p) for each trial
Specify the number of trials (n) you want to calculate probabilities for
Click "Calculate" to see the probability and expected value
View the probability mass function chart

Note: The calculator will show probabilities for values of n from 0 to the maximum number of trials you specify.

Formula and Assumptions

The probability mass function for the Negative Binomial Distribution is given by:

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

Where:

n = number of trials
k = number of successes
p = probability of success on each trial
C(n-1, k-1) = binomial coefficient

Assumptions

Trials are independent
Probability of success (p) is constant across trials
Only two possible outcomes for each trial (success/failure)

Limitations

The Negative Binomial Distribution assumes that the probability of success remains constant across trials. In real-world scenarios, this might not always be the case.

Worked Examples

Example 1: Quality Control

A manufacturer produces light bulbs with a known defect rate of 5%. What is the probability that exactly 3 defective bulbs will be found in the first 10 bulbs inspected?

Parameter	Value
Number of successes (k)	3
Probability of success (p)	0.05
Number of trials (n)	10

Using the formula:

P(X = 10) = C(9, 2) * (0.05)^3 * (0.95)^7 ≈ 0.034

Example 2: Sports Analytics

A basketball player has a free throw success rate of 70%. What is the probability that the player will need exactly 15 attempts to make 10 successful free throws?

Parameter	Value
Number of successes (k)	10
Probability of success (p)	0.7
Number of trials (n)	15

Using the formula:

P(X = 15) = C(14, 9) * (0.7)^10 * (0.3)^5 ≈ 0.087

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 would I use the Negative Binomial Distribution instead of the Poisson Distribution?: Use the Negative Binomial when you have over-dispersed data (variance greater than mean) and want to model the number of trials until a certain number of successes. Use Poisson when you have rare events and want to model the number of events in a fixed interval.
How do I interpret the probability mass function chart?: The chart shows the probability of achieving exactly k successes for different numbers of trials. The highest probability is typically near the expected value (k/p).
What are the practical applications of the Negative Binomial Distribution?: Common applications include quality control, reliability engineering, sports analytics, and any scenario where you're interested in the number of trials until a certain number of successes.
How does the Negative Binomial Distribution handle varying probabilities of success?: The standard Negative Binomial assumes a constant probability of success. For varying probabilities, you might need to use a more complex model like the Beta-Negative Binomial Distribution.