# 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 distributions to make sure you understand this topic correctly.

## What is the negative binomial distribution?

Both the binomial and negative binomial distributions involve consecutive events with a fixed probability of success.

For the binomial distribution, you determine the probability of a certain number of successes observed in $n$ trials. On the other hand, in negative binomial distributions, 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 all of them away. The probability that a passerby will take a leaflet from you is $p=0.4$.

- The probability of getting rid of $n$ leaflets if you try to hand them out to 50 people in binomial distribution is denoted by $P(n)$.
- 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 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 $X$); and
- 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 $Y$); and
- 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; and
- 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:

where:

- $P(Y=n)$ is the probability of the exact number of trials $n$ needed to achieve $r$ successes;
- $n$ is the total number of trials;
- $r$ is the number of successes;
- $p$ is the probability of one success; and
- ${n-1 \choose r-1}$ is the so-called "$n$ choose $r$" operation using the values $(n-1)$ and $(r-1)$. ${a \choose b}$ is the number of combinations of $b$ items that can possibly be assembled from a superset of $a$ items:

## How to use the Pascal distribution calculator

Let's solve the problem of the leaflets together.

- Determine the number of successes. $r=15$, because 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 is $p = 0.4$.
- Calculate the number of combinations ($n-1$ choose $r-1$, so $24$ choose $14$). You can use the combination calculator to do this. This number, in our case, is equal to $1,\!961,\!256$.
- Substitute all these values into the binomial probability formula above:

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