In the previous entry, we touched upon commonly occurring distributions: Bernoulli distribution, binomial distribution, uniform distribution, and normal distribution. Today, we’ll focus on binomial distribution and touch on the Bernoulli distribution, which is a special form of the former.

## What Is a Binomial Distribution?

A binomial distribution is a discrete probability distribution with the simplest possible structure behind it. It’s the simplest in the sense that each trial has exactly 2 possible outcomes: success or failure. The outcomes can be labeled as 1 or 0, yes or no, sale or no sale, or any random variable that captures the idea that there are 2 possible outcomes.

Take note, that the terms “success” and “failure” are non-judgemental terms. It’s only that the event of interest is deemed as “success” and the other is “failure.” If, for instance, the event of interest is defaulting on a loan—an event that is deemed negative in a social context—then defaulting on a loan is deemed as “success” in the equation. Basically, the outcome that you are calculating the possibility of happening is deemed the event of interest or the successful outcome.

## Bernoulli Distribution and Binomial Distribution: Similarities and Differences

As covered in the previous pose, the Bernoulli distribution, which is a common distribution. It is a special case of the binomial distribution in which only a single trial is done. That single trial corresponds to 2 outcomes (*x*) that can either be:

- 1, with probability
*p* - 0, with probability
*1 – p*

The symbol *p* represents the likelihood that the trial will end up with a success or that the event of interest will take place. The value of *p* is between 0 and 1 where 0 is representative of the event of interest not occurring and 1 is representative of the event of interest occurring. Since there are only 2 outcomes (success or failure) and the total probability for all possible outcomes must sum to 1. As such, if the probability of success is represented by *p*, then the probability of failure is be represented by *1 – p*.

The difference is that the Bernoulli distribution is concerned with a single experiment with only 2 possible outcomes. The binomial distribution, on the other hand, is concerned with the number of successes in a fixed number of independent Bernoulli trials. In summary:

*Bernoulli Distribution*

- Single Trial – It represents a single experiment with only two possible outcomes: success (1) or failure (0).
- One Probability – It has a single parameter (
*𝑝*), which represents the probability of success.

An example of this includes flipping a coin once and getting heads (1) or tails (0). Another would be asking a person if they have posted any photo on a specific social media platform (1) or not (0).

*Binomial Distribution*

- Multiple Trials – The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials.
- Two Parameters – It has 2 parameters:
*(𝑛)*or the number of trials, and*(𝑝)*or the probability of success in each trial.

In contrast to the previous examples, instead of flipping a coin once, using the binomial distribution means flipping a coin 10 times and counting the number of heads. Also, instead of asking a single person, the interviewer can ask 25 people if they have posted any photo on a social media platform and count the yes answers.

#### Assumptions When Carrying Out Binomial Distribution

When carrying out binomial distribution, the following are assumed to be true:

- The number of trials (
*n*) is fixed – Knowing the total number of trials is crucial because the distribution calculates the probability of achieving a certain number of successes out of these trials, such as getting 6 heads out of 10 coin flips. - Each trial is independent of other trials – This means that the probability of success remains constant across trials. If trials were dependent, the calculation of overall probabilities would be more complex and would not follow the binomial formula.
- There are only two possible outcomes for each trial: 1 and 0 – Again, the binomial distribution specifically models scenarios with binary outcomes such as yes or no, success or failure, and pass or fail type situations.
- The probability of success
*(p)*is the same for each trial – The binomial formula calculates the likelihood of a given number of successes based on a fixed success probability.

Do note that it’s still possible to use binomial distribution to calculate the probability of success when these assumptions are violated. However, the result of doing so will become approximate. This means that the result will not be strictly valid, but it can still prove to be useful in practical applications.

#### The Binomial Formula

The probability mass function (PMF) gives the probability that a discrete random variable is exactly equal to some value. This is calculated using the binomial formula:

Where:

*P(X = x)*is the probability of getting exactly*x*successes in*n*trials.*n*trials.*p*refers to the probability of success on a single trial.*(1 – p)*refers to the probability of failure on a single trial.

#### A Practical Demonstration of How to Apply Binomial Distributions

A practical example presents the best way to understand how binomial distributions can be relevant to real-life situations. Let’s say you are a business manager planning an email marketing campaign to promote a new product. From past campaigns, you know that the probability of a recipient opening an email (considered a “success”) is 20% ( *p* = 0.20). You plan to send this email to 1,000 customers ( *n* = 1,000). Applying binomial distributions can be done by following these steps:

*Define the Problem*

Identify the scenario where you want to apply the binomial distribution. Clearly define what constitutes a success and a failure.

For this particular situation, you want to find the probability that exactly 250 customers will open the email ( *x* = 250).

*Collect Data*

Gather relevant data to estimate the probability of success (*p*) for your specific situation. This could come from historical records, surveys, or pilot studies.

According to past campaigns, the probability that a recipient will open the email and count as a success for your email campaign is 20%.

*Determine Parameters *

Establish the number of trials (*n*) and the probability of success (*p*). For example, if you’re analyzing customer purchases, *n* could be the number of customers surveyed, and *p* could be the historical repeat purchase rate.

In the example, these refer to the number of trials (*n*), which is 1,000 emails, and probability of success (*p*), which is 0.20 (20%).

*Calculate Probabilities*

Use the binomial formula or statistical software to calculate the probabilities of different numbers of successes. This helps you understand the distribution of possible outcomes.

Let’s review the formula:

In this formula, *n*=1000, *x*=250, *p*=0.20, and *(1 – p)*=0.80

The binomial coefficient represents the number of ways to choose 250 successes from 1,000 trials. This can be calculated using:

Using a binomial calculator or software such as Excel, Python, or an online binomial calculator, you can input the values to find *P(X = 250)*.

*Interpret Results*

Analyze the probabilities to make informed decisions.

In this example, the calculated probability *P(X = 250)* is approximately 0.05. This means there is a 5% chance that exactly 250 out of 1,000 customers will open the email.

*Implement Strategies*

Based on your analysis, develop and implement strategies to optimize outcomes.

For this email campaign, if 250 opens are lower than expected, you might decide to improve your email content or target a larger audience. Alternatively, if the campaign performs better than predicted (e.g., 300 opens), it might indicate successful strategies worth repeating. At the same time, if the probability of getting fewer than 200 opens is high, you might need to have a contingency plan or follow up with additional marketing efforts.

#### The Relevance of Binomial Distribution in Business

Understanding and applying the binomial distribution can help you in various aspects of your business. For example, it can be quite useful in the following areas:

Suppose you want to predict how many customers out of a sample of 100 will make a repeat purchase if the probability of a repeat purchase is 30%. Using the binomial distribution, you can calculate the likelihood of different numbers of repeat purchases and use the answer to set realistic sales targets.*Customer Retention –*If you’re manufacturing products, you can use binomial distribution to predict the number of defective items in a batch. For example, if the defect rate is 2% and you produce 1000 items, you can determine the probability of having a certain number of defective items. This helps in maintaining quality standards and reducing waste.*Quality Control –*Let’s say that you are launching an email marketing campaign, and historical data shows that 20% of recipients typically respond. By applying the binomial distribution, you can predict the number of responses you might receive from a new campaign sent to 5000 customers.*Marketing Campaigns –***Risk Management –**When assessing business risks, you can use the binomial distribution to estimate the probability of certain events occurring, such as defaults on loans or failures in a system. This aids in developing strategies to mitigate these risks.

Incorporating binomial distributions into your decision-making process allows you to predict outcomes and manage uncertainties effectively. By understanding this statistical tool, you can enhance various aspects of your business operations.