**What is Binomial Distribution?**

The Binomial Distribution is a fundamental concept in statistics that describes **the number of successes in a fixed number of independent Bernoulli trials**.

Each trial has only two possible outcomes: success or failure.

**Bernoulli Trials**

A Bernoulli trial is a random experiment with exactly two possible outcomes: “success” and “failure”.

**Usages in Business**

It’s used in quality control (e.g., the probability of a batch having a certain number of defective items) and in finance (e.g., the likelihood of a certain number of successful investments).

**Formula of Binomial Distribution**

The formula for the Binomial Distribution is:

- is the probability of successes.
- is the number of trials.
- is the number of successes.
- is the probability of success on a single trial.

**Analysis with Binomial Distribution: Exercise**

Let’s delve deeper into our real-world scenario to understand how this works.

**Scenario**

A factory produces light bulbs, and historically, 5% of these bulbs are defective.

If a store buys **10 bulbs**, what’s the probability that **exactly 3 of them are defective?**

**Assumptions**:

- ( n = 10 ) (10 bulbs purchased)
- ( k = 3 ) (3 defective bulbs)
- ( p = 0.05 ) (5% defect rate)

**1. Combination Calculation**

First, we need to determine how many ways we can choose 3 defective bulbs out of 10.

This is a combination, represented as .

**2. Probability of 3 Defects**

Next, we’ll calculate the probability of getting 3 defective bulbs.

Since the defect rate is 5% or 0.05, the probability is .

**3. Probability of 7 Non-Defects**

We also need to consider the probability of the other 7 bulbs being non-defective.

This is .

**4. Combine Probabilities**

Now, we’ll combine all these probabilities using the Binomial Distribution formula:

So, if a store buys 10 bulbs from this factory, there’s a **1.04%** chance that exactly 3 of them will be defective.

This scenario can be visualized like this:

You can see the probabilities for each X ( number of defective bulbs ).