Binomial Distribution

By tossing your coin, either you have heads or tails. A condition that gives you only 2 results is said to be a Binomial Distribution. Also referred to as Binomial Probability Distribution, this mathematical concept has important applications in statics and many from probability theories. The popular ‘binomial test of statistical importance’ has the Binomial Probability Distribution as its core mathematical theory. Considering its significance from multiple points, we are going to learn all the important basics about Binomial Distribution with simple real-time examples.

A Brief Account of What is Binomial Distribution 

The results from a Binomial Probability Distribution will always have 2 outcomes only. So, there are 2 parameters to denote a Binomial condition.

  • “p” denotes the probability for any 1 event. 

  • “n” denotes the number of times an experiment or condition is done.

There is also this concept called the “Boolean-valued outcome”. The 2 outcomes for a question-like experiment is either YES or NO. As per the Boolean-value, the rate of success or failure for this condition can be denoted as 1/true/success/yes which is the binomial probability distribution of “p” and 0/false/failure/no can be represented with q = 1 − p. We have 3 more additional definitions to learn here as follows.

Bernoulli Trial - Getting either failed or success in an experiment Bernoulli Trial. This single-outcome result is also termed Bernoulli Experiment.

Bernoulli Distribution - To represent a single condition or experiment, the Bernoulli Distribution is preferred, where n=1.

Bernoulli Process - When there are more than 2 outcomes (series of results), then this sequencing is Bernoulli Process.

Defining Negative Binomial Probability Distribution

Consider the case of a discrete binomial probability distribution. The Bernoulli trials are identical but independent of each other. Before computing the failures (“r”), the total count of success that occurs first is called the Negative Binomial Probability Distribution. 

To brief the concept with a simple example, consider tossing a pair of coins. Let the outcomes 2 Heads be the success and all other results be a failure. So, the pair of coins is tossed for 5 times, until both the coins have heads on them. Now, the “r” in the condition is 5 (rate of failure) and all the remaining outcomes, i.e. the tosses that did not have 2 heads is the negative binomial distribution. 

Binomial Distribution from Real-Life Scenarios 

Here are a few real-life scenarios where a binomial distribution is applied.

  • Calculating the TRP of a Television channel, by taking a survey from households for whether they watch (YES) the channel or not (NO).

  • In the manufacturing of a commodity, estimating between the used and unused materials (raw).

  • 1 or 0 probability is the basis for finding the total count of votes for an electoral candidate. 

  • Counting the total number of female and male employees in an office setup.

The General Formula of Binomial Probability Distribution

Considering any random variable, the binomial distribution can be represented as given below:

P(x:n,p) = nCx px (1-p)n-x 


P(x:n,p) = nCx px (q)n-x

In the case of n-Bernoulli trials, the formula is written as follows:

P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x


“x” is any variable 0, 1, 2, 3…..

“n” denotes the total count of experiments

“q” gives the total  Failure Probability of a single experiment for 1 – p

“p” is the Probability rate of Success in a given experiment

Binomial Distribution and its 5 Major Properties

  1. Every single trial is an independent condition and so, this will not impact the outcome of 1 trial to that of another.

  2. For ‘n’ number of independent trials, only the total success is counted.

  3. The rate of failure and success will vary across every trial completed.

  4. Both these conditions are possible: “n” times repetition of trials and “n” count of independent trials 

  5. Only 2 outcomes occur: Yes/No. Success/Failure. 1/0. Yes/No.


Binomial Distribution can have only 2 outcomes. The characteristic of 1 will be positive and the other is negative. Statics and other mathematical fields make use of binomial probability distribution for finding the outcome for a set of independent experiments. The trial and outcomes vary across conditions. Having 2+ series of outcomes is said to be a  Bernoulli process while having n=1 is referred to as the Bernoulli Distribution. Bernoulli trial is nothing but getting either success or failure for a single experiment. Tossing a coin, rolling dice, writing an examination, counting the total number of votes, are some of the classic examples of Binomial Distribution. 

FAQ (Frequently Asked Questions)

1. What Does it Mean by a Hypergeometric Experiment?

Similar to the condition of a binomial distribution, the hypergeometric experiment is a statistical procedure where the sample size (n) is selected at random, without replacing anything from the given population. Here, ‘k’ denotes the success rate and failures are denoted as ‘n-k’.

2. Can you Calculate the Mean Value for an Experiment Using Binomial Distribution?

Yes. The mean value for an experiment could be calculated using a binomial distribution, by simply multiplying the total number of successes with the total count of trials.

3. State an Example for Binomial Distribution from the Medical Field.

Consider the disease “ABC”. Physicians are researching to find a drug for its treatment. Now, the total probability of the discovered drug effective for ABC has only 2 outcomes - the drug cures the disease (Success) or the drug does not cure the disease (Failure).

4. How is Binomial Distribution Useful to the Areas of Social Science?

In social science, Binomial Distribution plays a key role in the prediction of dichotomous outcome, to assess if the Democrat or the Republic will win the upcoming elections. Moreover, even important processes such as finding out the death rate and expected lifespan of an individual have its core model as Binomial Distribution.

5. Can 1 Trial have 2 Outcomes Simultaneously?

No. Every trial can have only 1 outcome. But the outcomes from multiple trials are exclusive and independent of each other. 

6. Will the Probability of Obtaining the Success Differ for Every New Trial Conducted?

No. Generally speaking, the success rate is constant for all the new trials that are being conducted. Also, every trial is independent of each other so one can expect success and failure interchangeably (fair chance of winning and losing).

7. Give any 1 Real-life Application for Binomial Distribution.

Consider the case of producing plant labels. The manufacturer will either label the product as ‘accepted’ or ‘defective’.