# Unlocking The Potential of Binomial Distribution

Normal distribution is obviously very close to all of our hearts, it’s widely used and has unprecedented power.

However, lurking in the shadows of normal distribution is the under-utilized, potential-filled **binomial distribution**. The most common example of this distribution is the classic coin example:

*“If we flip a fair coin with a fixed probability, and we flip the same coin **n** times, what is the probability of getting a certain number of heads?”*

In other words, if we flip a coin **n** number of times, the probability (**p**) of it landing on head or tails (**x**) follows a binomial distribution. The Probability Mass Function or *pmf* is a function of *n* trials with *p* probability of successes in those trials:

According to statistical topics covered by Yale University, binomial distribution should specifically be used to “summarize a group of independent observations by the number of observations in the group that represent one of two outcomes.” The single coin flip is an *ideal* example of an experiment with a binary outcome, however, the outcomes of the experiment don’t necessarily need to be equally likely in order for the experiment to be considered a “binomial.” For instance, an *unfair* coin is also an example of binomial distribution.

Let’s dive into some python code that runs the simulation of flipping a fair coin 20 times in 5000 trials. For this example, the following are our variables:

- n = 20
- p = 0.5; for a fair coin
- trials = 5000; repeat this experiment for this number of times

After 5000 trials, each tossing the coin 20 times, we were able to plot the histogram below. Look! A binomial distribution! How cool is it that the world works in patterns :)

To double check our results, I also defined a function that would plot the discrete *pmf* for given n trials (in our situation 20). Based on the plot below, we can see that the binomial distribution is the perfect model to simulate a coin toss. So in the future, we know that it’s unnecessary to sit there tossing a coin 5000 times …..cause I know we all have better things to do.

Conclusive remarks:

- Clearly, probability distributions do an incredible job of modelling real-life scenarios.
- Specifically in regards to binomial distribution examples, all we need are
**n**and**p**values to accurately model the outcomes. No need to sit there manually running 1000’s of trials! - Unlock the power of binomial distribution, normal distribution may be convenient but it’s time we harnessed the untapped potential that comes along with binomial distribution as well.