Bernoulli and binomial random variables are simple logical tests in a trial with two possible outcomes. Both of these random variable distributions are used closely in statistics for probability studies.

Let us discuss what Bernoulli and binomial random variables and their key differences are.

**What is a Bernoulli Random Variable?**

A Bernoulli variable is a type of random variable that can take only one of the two possible values. It means there are only two possible results in a Bernoulli random distribution.

The term comes from the Bernoulli Trial which means having only one trial and two possible expected results for it. Bernoulli random variables and Indicator Variables are closely linked concepts.

It can also be defined as a discrete random variable that has finite values. The distribution will have only one trial and one result at a time.

If a random variable has more than two values and it needs more than one trial, it does not fall under the scope of the Bernoulli distribution.

**Understanding Bernoulli Distribution**

A Bernoulli trial is expressed as “success or failure”. Its results are expressed as “0 or 1”, “true or false”, “yes or no”, “pass or fail”, and so on.

Each trial conducted must be independent of the other one. Each time the trial is conducted, the two possible outcomes must have equal outcome chances.

Bernoulli distribution is a concept linked to discrete distributions. It means each trial conducted in the distribution is discrete (independent) of other trials.

Bernoulli random variables are used in regression and economic analysis to find a relationship between two variables or find out the probability of an outcome as compared to the other possible result.

**Formula**

Any alphabet can denote the Bernoulli random variable. Usually, it is expressed as (p).

So, if we want to know whether a variable “X” is a Bernoulli Random Variable with possible outcomes “1 or 0”, then:

The Probability mass function can also be written as:

P (X = 1) = p

P (X = 0) = (1 − p)

If we assume the possible outcome is denoted by “n”, then:

P (n; X) = X^{n }(1 – X) ^{1-n}

**Example**

Some common examples of Bernoulli distributions include conducting a “yes, no” trial, a coin flip, a pass-fail result, and so on.

For instance, if we flip a coin, it can fall to heads or tails every time. There are only two possible outcomes and each trial does not impact the other one.

Similarly, if a student sits an exam, the result can either be a pass or fail provided a certain criterion is set. Then, the next exam result will have the same probability and will not depend on the previous one.

Bernoulli distribution can also be used in economics, regression, and medical tests to conduct a simple “yes, no” test. However, each test must comprise of a single trial and the result should have only two possibilities.

**What is a Binomial Random Variable?**

A binomial random variable is a discrete variable representing the collection of successive independently successful trials in a Bernoulli distribution.

A binomial distribution is a collection of successful trials in a Bernoulli distribution. Thus, it will possess the same characteristics but will have more than one trial for the experiment.

The underlying concepts of a binomial distribution are to have two possible outcomes of an experiment (or trial) and there is more than one trial in the distribution.

Thus, if we repeat the Bernoulli distribution trials successively, we’ll form the binomial distribution trend. Therefore, it is an extension of discrete random variables defined in the Bernoulli distribution theory.

**Understanding Binomial Distribution**

We can fully understand the concept of a binomial random variable (and distribution) by understanding the major characteristics of the theory.

**Trials Should be Independent**

This condition simply means that the successive trials conducted in the distribution should not affect each other.

So, each trial will offer a unique result regardless of the previous trial’s result. It can be achieved by increasing the number of trials and setting unique outcomes.

**A Set Number of Trials**

Although you need repeated and successive trials of Bernoulli random variables, you’ll need to set a fixed number of trials first.

It means the binomial distribution should have a finite range. In other words, conducting trials should reveal measurable results.

Each trial should be conducted in similar circumstances and the method should be the same for all experiments.

**Predefined Probability of Outcomes**

The population data in the trial should hold such characteristics that the probability of outcomes remains fixed for all trials.

It is mainly achievable when there are only two possible outcomes. For example, if a student takes exams, the result can be either pass or fail each time and the chances of each occurrence will be the same as well.

**Outcomes are Mutually Exclusive**

It simply means that if you yield one result, the other is impossible. So, the outcomes of the trial remain mutually exclusive of each other.

For instance, a coin flip can have only one result at a time and the second one is not possible at the same time.

**Formula**

Suppose we denote the Binomial random variable by X. If the number of trials is “n” and the number of successes is “k”, then the probability of the binomial random variable can be given as:

Here the value of k = 1,2,3, …. n.

**Example**

Suppose we want to diagnose an error in a signal network system. A new network code allows a 4-bit code to be encoded as 7-bit. The system can reconstruct the message if 0 or 1 bits are lost.

The probability of a bit being lost is 0.1. How will the new network code change the reliability of the system?

Solution:

Let’s suppose “X” is the binomial random variable Bin (7, 0.1).

Then,

Without using the error correcting new code then:

So, using the new error-correcting code improves the system reliability by around 30% roughly.

**Bernoulli and Binomial Random Variables – Key Differences**

As we have discussed above, both types of random variables work when there are two possible trial outcomes.

The Bernoulli distribution occurs when only one trial is required for a test and has two probable outcomes with a fixed outcome probability.

The binomial distribution occurs when there is more than one successive trial.

Trials in both cases should be independent and come with a fixed number. The outcome probabilities should also be fixed.