Course Content

# Probability Theory Basics

4. Commonly Used Continuous Distributions

5. Covariance and Correlation

Probability Theory Basics

## Law of Total Probability

The **law of total probability** is a fundamental concept in probability theory. This law can be formulated as follows:

Let's provide some explanations:

- We have split our space of elementary events into n different incompatible events.
- We want to calculate the probability of some other event in this space of elementary events.
- We can calculate P(A) using the formula described above.

This law is often used when a stochastic experiment can be divided into different stages, and each stage is stochastic too.

Let's consider an example involving a manufacturing company that produces two types of products: **Product 1** and **Product 2**. The company produces `60%`

of **Product 1** and `40%`

of **Product 2**. The defect rate for Product 1 is `10%`

, while the defect rate for Product 2 is `5%`

.
We want to calculate the probability of randomly selecting a defective product from the company's inventory.

In this example:

Event A: Selecting a defective product.

Partition events: H₁ = Selecting Product 1, H₂ = Selecting Product 2.

Now we can use the law of total probability to solve this task:

You have two baskets: the first one contains 3 cats' toys and 7 dog's (10 toys), the second one contains 12 cat's toys and 8 dog's (20 toys). The probability to choose the first basket is 0.4, and to choose second is 0.6. Calculate the probability to get cats' toy.

Select the correct answer

Everything was clear?