In probability theory, independence and incompatibility are concepts related to the relationship between random events.

1. **Independence**: Two events are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the occurrence or non-occurrence of the other event. In other words, knowing whether one event happens provides no information about the likelihood of the other event happening.   
**Events A and B are independent if P(A intersection B) = P(A)\*P(B)**;
2. **Incompatibility**: Two events are incompatible if they cannot occur simultaneously. If the occurrence of one event excludes the possibility of the other event happening, they are considered incompatible. For example, flipping a coin and getting heads and tails simultaneously is incompatible since the coin can only show one side at a time.   
**Events A and B are incompatible if P(A intersection B) = 0**.

Examples of independent and incompatible events: 

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      <th><strong>Independent events</strong></th>
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      <td>Results of flipping a fair coin and rolling a fair die</td>
      <td>Rolling a die and getting an odd number and an even number on the same roll</td>
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      <td>Selecting a card from a deck and getting a red card and a black card at the same time</td>
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You draw a card from a standard deck with replacement (after we have drawn a card, we return it back to the deck) . What is the probability of drawing a red card (heart or diamond) followed by drawing a black card (spade or club)?

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

Independence and Incompatibility of Random Events