A Bernoulli Distribution is the probability distribution of a random variable which takes the value 1 with probability p and value 0 with probability 1 – p, i.e.

$$
begin{cases}
1-p & text{for} k=0
p & text{for} k=1
end{cases}$$

We will use the example of left-handedness. Approximately 10% of the population are left-handed (p=0.1).

We want to know, out of a random sample of 10 people, what is the probability of 3 of these 10 people being left handed?

We assign a 1 to each person if they are left handed and 0 otherwise:

Bernoulli’s principle formulated by Daniel Bernoulli states that as the speed of a moving fluid increases (liquid or gas), the pressure within the fluid decreases. Although Bernoulli deduced the law, it was Leonhard Euler who derived Bernoulli’s equation in its usual form in the year 1752. The Bernoulli trials process, named after Jacob Bernoulli, is one of the simplest yet most important random processes in probability. Essentially, the process is the mathematical abstraction of coin tossing, but because of its wide applicability, it is usually stated in terms of a sequence of generic trials. Bernoulli and Binomial Page 12 of 26 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ Synthesis 6. The Bernoulli Distribution The Bernoulli Distribution is an example of a discrete probability distribution. It is often the. European version of roulette with 37 numbers (with a single zero). However, the results can be.,M is a random variable with a Bernoulli distribution with (0) n.

$P(X=1) = 0.1$
$P(X=0) = 0.9$

A Binomial distribution is derived from the Bernoulli distribution.

We’ll start with the simpler problem:

What is the probability of the first 3 people we pick being left-handed, followed by 7 people being right-handed?

This is just $ 0.1 ^3 times 0.9 ^7$

What if we wanted the last 3 people to be left-handed?

This is just $0.9^7 times 0.1^3$, the same answer.

In fact, no matter how we arrange the 3 people, we will always end up with the same probability ($ 4.7 times 10^{-4} $).

So we have to add up all the ways we can arrange the 3 people being picked.

There are $10!$ ways to arrange 10 people and there are $3!$ ways to arrange the 3 people that are picked and $7!$ ways to arrange the 7 people that aren’t picked.

This is given as:
$$dfrac{10!}{3! 7!}$$

Or more commonly, “10 choose 3”. The “n choose k” notation is written as:
$$
begin{equation*}
binom{n}{k}
end{equation*} = dfrac{n!}{k! (n-k)!}
$$

We can now caclulate the probability that there are 3 left-handed people in a random selection of 10 people as:

$$
P(X=3) = begin{equation*}
binom{10}{3}
end{equation*} (0.1)^3 (0.9)^7
$$

This will generalise such that:

$$
P(X=k) = begin{equation*}
binom{n}{k}
end{equation*} (p)^k (1-p)^{n-k}
$$

Scipy’s stats package has a binomial package that can be used to calculate these probabilities:

We can use this function to calculate what the probability of 3 or fewer people being left-handed from a selection of 10 people.

$$
P(X leq 3) = sum_{i=0}^{3} begin{equation*}
binom{10}{i}
end{equation*} (0.1)^i (0.9)^{n-i}
$$

Or we could plot our probability results for each value up to all 10 people being left-handed:

We can see there is almost negligible chance of getting more than 6 left-handed people in a random group of 10 people.

Roulette

On an American roulette wheel there are 38 squares:

18 black
18 red
2 green

We bet on black 10 times in a row, what are the chances of winning more than half of these?

$$
P(X gt 5) = sum_{i=6}^{10} begin{equation*}
binom{10}{i}
end{equation*} bigg(dfrac{18}{38}bigg)^i bigg(1-dfrac{18}{38}bigg)^{n-i}
$$

A Poisson distribution is a limiting version of the binomial distribution, where $n$ becomes large and $np$ approaches some value $lambda$, which is the mean value.

The Poisson distribution can be used for the number of events in other specified intervals such as distance, area or volume. Examples that may follow a Poisson include the number of phone calls received by a call center per hour and the number of decay events per second from a radioactive source.

It is calculated as:

$ P(k) = e^{-lambda} dfrac{lambda^k}{k!} $

The average number of goals in a World Cup football match is 2.5.

We would like to know the probability of 4 goals in a match.

Again, scipy has in-built functions for calculating this and we can use this to calculate the probability of any number of goals in a World Cup match.

In today’s post, I’m going to give you intuition about the Bernoulli distribution. This is one of the simplest and yet most famous discrete probability distributions. Not only that, it is the basis of many other more complex distributions.

This post is part of my series on discrete probability distributions.

In my introductory post on probability distributions I gave you some intuition about the two main types: discrete and continuous. And in a follow-up post, I showed you the general formulas for calculating the mean and variance of any probability distribution.

The goal of this post is to give you more details and intuition about the most famous of all discrete probability distributions, including its probability mass function, mean, and variance.

Introduction

In the overview post on discrete probability distributions, I showed you that you can see the main distinction between discrete and continuous probability distributions in the sample spaces associated with them.

Roulette Bernoulli Model

In short, the sample space of a discrete probability distribution is either finite or countably infinite. On the other hand, the sample space of a continuous probability distribution is uncountably infinite. Therefore, you can think of discrete sample spaces as subsets of natural numbers and of continuous sample spaces as subsets of real numbers. You can read more about this in the section discussion discrete sample spaces.

I also told you that every discrete probability distribution is actually a class of specific distributions defined by a probability mass function (PMF). The specific distributions are associated with specific values of one or more parameters of the PMF. You can read more about this in the parameters section of the same post.

The Bernoulli distribution deals with random variables that have exactly 2 possible outcomes. And it simply assigns a probability to each of those outcomes. Pretty simply, isn’t it?

A single realization of a Bernoulli random variable is called a Bernoulli trial. On the other hand, a sequence of realizations is called a Bernoulli sequence or, more formally, a Bernoulli process. Different types of Bernoulli sequences give rise to more complicated distributions, like the binomial distribution and the Poisson distribution.

Before I show you the details of the Bernoulli distribution, let me tell you a few words about its name.

History of the Bernoulli distribution

The name of this distribution and all related terms comes from the 17th century Swiss mathematician Jacob Bernoulli. Now, he wasn’t really the one who coin the term itself. At the time the concept of a probability distribution hadn’t even been discovered yet.

The distribution is named after Bernoulli because he was the one who explicitly defined what we today call a Bernoulli trial. Namely, an experiment with only two possible outcomes. For convenience, these outcomes are usually called “success” and “failure” (but don’t read too much into these labels). This tradition started from Bernoulli himself in his book Ars Conjectandi (“The Art of Conjecturing”), published 8 years after his death. Take a look at this quote:

To avoid tedious circumlocution, I will call the cases in which a certain event can happen fecund or fertile. I will call sterile those cases in which the event can not happen. I will also call experiments fecund or fertile in which one of the fertile cases is discovered to occur; and I will call nonfecund or sterile those in which one of the sterile cases is observed to happen.

Many historians consider Ars Conjectandi the f