This was a good year for me workwise. It was my long-time dream to start my own business. I started a small weekend only fashion store out of my garage three months back. I managed to identify a supplier who shares my taste and just to make sure, I sent out samples to my friends and got their opinion as well. Now I’ve opened shop and my business hours are 9 AM to 11 PM every weekend.

I am the sales girl/cashier and just before my customers exit, I have a tablet set up that lets them give me a super quick feedback on my collection. It just displays the question “Hello! Did you like the collection that you just saw?” If they did, they will press the happy green face and if they didn’t they pressed the angry red face.

I have the data on my customer reviews and I want to find out the probability of me getting a certain number of happy faces next week so I can order accordingly from my supplier and maintain or improve my standards of service. The probability of me getting angry faces is also important so I can improve the collection or the service I provide. To gain these insights I am using the binomial distribution.

Please download the workbook from the link below to gain access to the entire work.

Binomial Distribution Workbook

### What is a Binomial Distribution?

The word ‘binomial’ means consisting of two terms. We use this method in trials to find the probability of outcomes which are Boolean-valued or true-false based. It can have only one outcome at a time. Such trials are called Bernoulli trials, named after Jacob Bernoulli who made important contributions to the field of probability.

Since we have many instances in our daily life and in our work that can be simplified into a yes or no answer, we can use this method for finding probability extensively.

In order to use binomial distribution, the following has to be true.

- The number of trials should be fixed.
- Trials should be independent of each other.
- Trials should have 2 outcomes in the true/false format.
- The probability of these outcomes should be the same throughout the experiment.

Whenever we calculate the probability, we either need to collect information about the experiment either through repetition or with the help of past information or both.

We calculate binomial distribution using the following formula.

b(x; n, P) = nCx * Px * (1 – P)n – x

n is the number of Bernoulli trials

x is the total number of successes

p is the probability of success in a single trial (Which we already know of)

q is the probability of failure in a single trial (1-p)

### Coming Back to My Store…

Now that we have some idea about the binomial distribution, let us put it to good use. Following is a subset of customer reviews I collected over 10 days. Download workbook here or from the top of the page.

From this data, let us get the information we need. We need to calculate the probability of getting a certain number unhappy customers *a day*. We can also calculate the probability that the number of unhappy customers falls within a range. For example, what is the probability that out of all the customers who come, the number of unhappy customers will range from 5 to 10?Based on this information, we can do further investigation and improve our service.

First, let us derive some basic information from the data itself.

Total number of customers = 732

Number of happy customers = 617

Number of unhappy customers = 115

Revisiting our binomial equation, we have,

b(x; n, P) = nCx * Px * (1 – P)n – x

We need to find the probability of favorable outcomes and we need the number of trials we conduct per day. We also need to decide the number of unhappy customers we are interested in looking at, x.

We can calculate the probability of happy and unhappy customers coming into our store.

Probability of happy customers = 617 / 732 = 84%

Probability of happy customers = 115 / 732 = 16%

We have reviews of customers walking in and out for 10 days. Let us consider its average as the number of trials per day.

Average number of customers per day = 73

So, what is the probability that exactly 15 unhappy customers leaving my store next time?

P(Unhappy Customers = 15) = 6%

This is the probability mass function of binomial distribution. It gives you the probability of getting exactly the value we are interested in.

This information does not say much, does it? I want to see what is the probability that up to 15 unhappy customers walk out of my store next time.

P(Unhappy Customers < = 15) = 90%

Now this says something, doesn’t it? Around 20% of my customers are unhappy with what they see.

Now I want to see if I can put the number of unhappy customers between 5 and 20.

P(Unhappy Customers 5<=X<=20) =99%

Based on the information I see, if I need to scale up my business, I need to fix this problem. From next week on, I need to improve my collection and service so that I can reduce the number of unhappy customers.

# Road Ahead

Binomial distribution can be applied in our business applications to assess risks, manage inventory, manage retention rates etc. Based on the results from a fixed set of trials, we can foresee the possibility of the number of successes or failures and manage life accordingly. How powerful is that?

What are some of the Bernoulli trials in your life? Get that data and go ahead and predict.

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