**The Bernoulli distribution is a theoretical model used to represent a discrete ****random variable**** which can only result in two mutually exclusive events. **

In other words, Bernoulli’s distribution is a distribution applied to a discrete random variable, which can only result in two possible events: “success” and “no success”.

Recommended articles: **sample space** , Bernoulli distribution example and Laplace Rule.

## Bernoulli experiments

An experiment is a random action which we have no way of predicting, such as the result of throwing a dice. In the Bernoulli distribution we only do a **single experiment** . In the case that more than one experiment is performed, as in the **Binomial distribution** , the experiments are independent of each other.

## “Success” and “and not success”

They are experiments where the final situation can only result in two exclusive results or events:

- The result we expect to happen. That is, ”
**success**.” - The result other than the result we expect to occur. That is, ”
**no success**.”

## P parameter

Given a discrete random variable Z whose frequency can satisfactorily approximate a Bernoulli distribution with a parameter p.

The frequency of the random variable Z can be approximated satisfactorily by a Bernoulli distribution with probability p.

Generally the parameter p is used to indicate the probability of success of the discrete random variable Z. Then:

Possible results of the random variable Z.

- If the random variable Z results in the result that we had defined as “success” at the beginning of the experiment, (Z = 1), then the probability of obtaining that particular result is (p).
- If the variable Z results in a different result than the one we had defined as “not successful” at the beginning of the experiment, (Z = 0), then the probability of obtaining that specific result is (1-p).

## Important

It is important to note that the result ” **not successful** ” does not refer to the opposite of “success”, but refers to any case **other** than the one that represents “success” as long as there are more than two possibilities.

That is, in the case of rolling a dice, if the variable “success” refers to obtaining a four (4) in a roll, the variable “not success” will be any result other than four (4) that we can obtain in a roll. lying

**Sample space** : {1,2,3,4,5,6}.

In the case of a coin (not tricked), we can only get two possible results: face or cross. Then, in this case the variable “no success” will be effectively the opposite of the variable “success”.

Sample space: {1,2}.

## Parameter formula p and the Laplace Rule:

To obtain the p parameter we use the Laplace Rule:

Laplace rule.

**Possible cases:**These are all possible results that we can obtain in an experiment. For example, if the experiment is to roll a dice, we will have six (6) possible cases because a dice has only six (6) faces.**Probable cases**: These are the results that appear in each experiment**sequentially**, that is, the results are**exclusive**: if one result occurs, the others cannot occur. In the experiment of throwing a dice, each face of the dice is a probable case. In other words, a two (2) or a five (5) result are examples of probable cases in the experiment of throwing a dice.