The binomial distribution and the negative binomial distribution are both discrete probability distributions used to model the probability of success in a sequence of independent and identically distributed Bernoulli trials. Both distributions are characterized by the probability of success (p) and the number of trials (n). However, there are some key differences between the two distributions:

## Binomial Distribution:

The binomial distribution is a probability distribution that describes the **number of successes** in a series of independent and identically distributed Bernoulli trials. In other words, it is a distribution that describes the number of times an event will occur in a fixed number of trials, where the probability of success on each trial is constant. The binomial distribution is a discrete distribution, which means that it can take on only a finite or countably infinite number of values.

Here are an example of using the binomial distributions in the context of flipping a coin:

- Modelling the probability of getting exactly 4 heads in 10 coin flips

## Negative Binomial Distribution:

The negative binomial distribution is a probability distribution that describes the **number of failures that must be experienced before a certain number of successes **occurs in a series of independent and identically distributed Bernoulli trials.

Here are an example of using the negative binomial distributions in the context of flipping a coin:

- Modelling the probability of getting 3 heads before getting 2 tails in a coin-flipping experiment

The binomial distribution is used to model the probability of getting a certain number of heads in a given number of coin flips, while the negative binomial distribution is used to model the probability of getting a certain number of heads before a certain number of tails in a coin flipping experiment.

## Summary:

In summary, the binomial distribution models the number of successes in a given number of trials, while the negative binomial distribution models the number of failures before a given number of successes.