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

Let's provide some explanations:
1. We have **split** our space of elementary events into `n` different **incompatible events**;
2. We want to calculate the probability of **some other event** in this space of elementary events;
3. 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. 
## Example
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:

# Probability of selecting Product 1 and Product 2
P_H1 = 0.6
P_H2 = 0.4

# Defect rates for Product 1 and Product 2
P_A_cond_H1 = 0.1
P_A_cond_H2 = 0.05

# Calculate the overall probability of selecting a defective product
P_A = P_A_cond_H1 * P_H1 + P_A_cond_H2 * P_H2

# Print the results
print(f'The overall probability of selecting a defective product is {P_A:.4f}')

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. 

Probability theory is a fundamental branch of mathematics that deals with the study of uncertainty and randomness. It provides a framework for understanding and quantifying uncertainty in various fields, including statistics, data analysis, machine learning, finance, physics, and more.

We will start our way of learning probability theory by considering some basic definitions and rules: what is a stochastic experiment and random event, what is independence and incompatibility of events in the context of probability theory, what is the probability and how can we calculate probabilities of different elementary events. 

In real-life tasks, we often have to deal with complex relationships and, as a result, calculate probabilities of several events or events that depend on each other. Let's consider how we can do this using probability theory.

To solve many real problems in probability theory, special models have been created that describe a particular situation. Let's consider some of the most used models that can be used to describe some discrete results of stochastic experiments.

What if the result of a stochastic experiment cannot be described by a discrete value? For this, models that work with continuous values are used. Consider the most popular of these models.

Often we are faced with the task of checking the dependence of the results of different stochastic experiments on each other. Moreover, it is necessary not only to assess the presence of dependencies but also to somehow quantify the degree of dependencies. To solve these problems, we can use covariance and correlation.

Probability Theory Basics

Law of Total Probability

Example