What is binomial distribution? Its Formulas & Examples

what is binomial distribution

What Is Binomial Distribution 

Several students face difficulty while solving the problems of the binomial distribution, but there might be various reasons for this, such as students are not able to understand the term what is binomial distribution? How to use its formula? Therefore, this post helps those students who are confused with such questions as it has all the details about:

  • Binomial distribution definition?
  • The Binomial distribution formula. 
  • Binomial distribution examples. 

So let’s discuss all these terms step by step in the upcoming paragraphs. 

What is binomial distribution?

It is a probability distribution of success or failure results in a survey or an experiment that might be used several times. That has two possible results. For instance, a coin is tossed that has two possible results: tails or heads. And the test could be resulted as pass or fail. 

It is a discrete distribution that is used in statistics that opposes a continuous distribution. The reason for this is that it only counts two states. That are represented as 0 (for failure) or 1 (for success) for a provided number of experiments. Therefore, it donates the probability for x successes in several trials n, giving the probability p of successive trails.

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This is the binomial distribution definition that helps you to understand the meaning of the binomial distribution now, we will discuss the criteria of it.

Criteria of binomial distribution

The criteria of the binomial distribution need to satisfy these three conditions:

  • The number of trials or observation must be fixed: If you have a certain number of the trial. Then you can easily find out the probability of it. For example, if you throw a coin, then the probability of coming a head is 50%. But if you throw the coin almost 10 times. Then the probability of coming head is also increased up to almost 100%. 
  • The success’ probability is the same as for another trial too.
  • Each trial or observation must be independent: This means that any of the experiments do not have an effect on another or next trail.

Before proceeding to the further details, let’s discuss the Bernoulli Distribution and how these terms related to each other. 

What is Bernoulli Distribution?

As per the Washington State University, “If every Bernoulli experiment is independent, the successive no. in Bernoulli experiment has a binomial distribution. In other words, the Bernoulli distribution is the binomial distribution that has a value of n=1.”

The Bernoulli distribution is the set of the Bernoulli experiment. Every trial has a possible result, selected from S (for success), F (for failure), and each trial’s probability would be the same. The probability of success is 1 minus the probability of failure that is P(S) = 1- p. Finally, all Bernoulli experiment is independent of each other, and the success’ probability does not alter from experiment to experiment. 

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The real examples of what is binomial distributions

Various examples are based on real-life. For instance: If a new medicine is launched to cure a particular disease. So there is the possibility of success and failure. 

Binomial distribution formula:

When you know about what is binomial distribution, let’s get the details about it:

b(x; n, P) = nCx * Px * (1 – P)n – x


b = binomial probability

x = total number of “successes” (fail or pass, tails or heads, etc.)

P = probability of success on an individual experiment

n = number of experiment

There is another formula to write it that is a slightly different way that is:


Binomial distribution examples:

Now, we will describe the way to use the it. We use it to solve the different mathematics problems:

Example 1:

A coin is thrown 5 times. What is binomial distribution of coming exactly 3 heads?

We are using the formula: b(x; n, P) – nCx * Px * (1 – P)n – x

The no. of experients (n) is 5

The success’ odds (“getting a heads”) is 0.5 (therefore 1-p = 0.5)

x = 3

P(x=3) = 5C3 * 0.5^3 * 0.5^2 = 10*0.125 * 0.25 = 0.3125.

Example 2:

80% of persons who buy health insurance are men.  If 6 health insurance buyers are chosen randomly. Find the binomial distribution that exactly 3 are men.

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1st Step: Recognize ‘n’ from the question. In our binomial example 2, n (the number of chosen items randomly) is 6.

2nd Step: Find ‘X’ from the question. X is 3.

3rd Step: Solve the first portion of the formula. The first portion of the binomial distribution formula is

n! / (n – X)!  X!

Put the values of each: 

6! / ((6 – 3)! × 3!)

That is equal to 40. Now let’s proceed to further discussion. 

4th Step: Solve the value of p and q. p is the success’ probability, and q is the failure’s probability. We have the value of p = 80%, or .8. So the failure’s probability is 1 – .8 = .2 (20%).

5th Step: Now proceed to the second portion of the formula.


= .8^3

= 0.512

6th Step: Solve the next portion of the formula.

q(n – X)

= .2(6-3)

= .2^3

= .008

7th Step: Multiply the value from step 3, 5, and 6 together to get the desired answer.

40 × .512 × .008 = 0.16384.


The above-mentioned details will help you to solve the problems of it as this post have all the relevant information about what is binomial distribution, its formula, and examples to use those formulas. Still, find any difficulty?

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