# Negative Binomial Distribution Calculator

This negative binomial distribution calculator, otherwise called the Pascal distribution calculator, can help you determine what is the probability of requiring n trials to achieve a fixed number of successes. We will also provide you with a list of examples of negative binomial distribution to make sure you understand this topic correctly.

## What is the negative binomial distribution?

Both the binomial and negative binomial distribution concern consecutive events with fixed probability of success.

In the binomial distribution, you determine the probability of a certain number of successes observed in n trials. In negative binomial distributions, on the other hand, your random variable is the number of trials needed to get r successes.

Picture it like this: you are handing out leaflets on a street. You have 15 leaflets on you and cannot stop until you give away all of them. The probability that a passerby will take a leaflet from you is 0.4.

- The probability of getting rid of
**n**leaflets if you try to hand them out to 50 people in binomial distribution is**P(n)**. - The probability of having to try
**m**times to give out a leaflet to get rid of all 15 of them is a negative binomial distribution in**P(m)**.

Notice that while the binomial distribution allows you to try only a specific number of times, the negative has a tail at infinity. After all, you might just have a terrible luck and never give out the last leaflet.

To sum up, in a binomial distribution, you have:

- Fixed number of trials (n);
- Random variable the number of successes (denoted with X);
- Possible values are 0 β€ X β€ n.

In the negative binomial distribution, you have:

- Fixed number of successes (r);
- Random variable the number of trials until the r-th success (denoted with Y);
- Possible values are Y β₯ r.

## Negative binomial distribution examples

Some examples of negative binomial distribution include:

- How many times you need to roll a die until you get three results of 6;
- How many times you need to knock on doors during the Halloween night until you collect 20 candy bars;
- How many times you need to flip a coin to get four heads;
- How many attempts you need to score three goals in a match.

## The negative binomial distribution formula

Our negative binomial calculator uses the following formula:

`P(Y=n) = (n-1)C(r-1) * p^r * (1-p)^(n-r)`

where:

**n**is the total number of trials;**r**is the number of successes;**p**is the probability of one success;**(n-1)C(r-1)**is the number of combinations (so-called "n choose r"), using the values (n-1) and (r-1);**P(Y=n)**is the probability of the exact number of trials**n**needed to achieve**r**successes.

## How to use the Pascal distribution calculator

Let's solve the problem of the leaflets together.

- Determine the number of successes. r is equal to 15, as we have to hand out 15 leaflets.
- Determine the number of trials. Let's say you want to calculate the probability of handing all leaflets out in 25 trials.
- The probability of a stranger taking a leaflet from you is equal to 0.4. Hence, the probability of an individual success p = 0.4.
- Calculate the number of combinations (n-1 choose r-1, so 24 choose 14). You can use the combination calculator to do it. This number in our case is equal to 1,961,256.
- Substitute all these values into the binomial probability formula above:

`P(Y=50) = 1,961,256 * 0.4^15 * (1-0.4)^(25-15) = 0.01273`

You can also save yourself some time and use the negative binomial distribution calculator instead :)