**What is Probability Distribution?**

A probability distribution is a statistical function that describes the likelihood of obtaining possible outcomes in an experiment.

In simpler terms, **it tells us how probable different outcomes are in a random experiment**.

**Example**

**Think of rolling a fair six-sided dice**. The probability of any one side landing face up is .

A probability distribution would list each side of the dice and its associated probability.

**Mathematical Notation**

For a discrete random variable X:

- = Probability
- = Random variable
- = Specific outcome

**Usages of Probability Distribution in Business**

Probability distributions play a pivotal role in various business sectors:

**Finance**: To model and predict stock prices, interest rates, and other financial metrics.**Marketing**: To forecast sales, customer behavior, and market trends.**Supply Chain**: To predict product demand and optimize inventory levels.

**Types of Probability Distributions**

There are several types of probability distributions, each with its characteristics and formulas:

**Uniform Distribution**

All outcomes are equally likely.

**Formula**:

Where is the number of possible outcomes.

**Binomial Distribution**

**Binaomial Distribution** is commonly used distribution, representing the number of successes in a fixed number of Bernoulli trials.

**Formula**:

= Number of trials

= Number of successes

= Probability of success on a single trial

**Normal Distribution**

Also known as **the bell curve**, Normal Distribution describes continuous data that clusters around the mean.

**Formula**:

= Mean

= Standard deviation

**Predicting with Probability Distribution**

Probability distributions are not just theoretical constructs; they have practical applications, especially in **predicting outcomes based on historical data**.

Let’s delve deeper into our ice cream shop example:

**Imagine you run a small ice cream shop**. Over the past month, you’ve kept track of your ice cream sales and found the following distribution of sales probabilities:

- Vanilla: 30%
- Chocolate: 40%
- Strawberry: 30%

Now, let’s say you’re expecting** 100 customers **tomorrow.

**Predicting Sales**

**Vanilla**

customers

We expect to see 30 customers purchasing the vanilla icecreams.

**Chocolate**

customers

We expect to see 40 customers purchasing the chocolate icecreams.

**Strawberry**

We expect to see 30 customers purchasing the strawberry icecreams.

**Stocking Up**

Based on the prediction, you’d prepare ingredients and stock up more on chocolate since it’s expected to be the most popular flavor for the day.

**Risk Management**

Probability distributions also help in assessing risks.

For instance, if there’s a sudden shortage of chocolate, you can gauge the impact on sales and quickly pivot to promoting vanilla and strawberry flavors.

**Refining Predictions**

As you gather more sales data over time, you can refine your probability distribution, making your predictions more accurate.

Maybe during summer, the preference shifts more towards strawberry, and you can adjust accordingly.