We have the following properties of the probability function:

      The previous properties represent formulas currently used in probability calculus on a finite field of events.
      Property (P9) is the main calculus formula for applications in finite cases.
      In addition, if  {Ω, Σ, P} is a σ-field, we also have the following properties:

Independent events. Conditional probability

Let us consider the experiment of tossing two coins and let A – heads on first coin and  B – heads on second coin be two events. The occurrence of event A and its probability do not depend on the occurrence of event B, and vice versa. In this case, events A and B are said to be independent (each isindependent of the other).

According to this definition, in the previous example we have: P(A and B) = P(A) x P(B) = (1/2) x (1/2) = 1/4.

Consider an urn containing four white balls and three black balls. Two people extract one ball each from the urn. Let A – first person is extracting a white ball and B – second person is extracting a white ball be two events. The probability of event B, in the absence of information about A, is 4/7. If event A has occurred, the probability of event B is 1/2, so event B depends on event A. Therefore, these two events are not independent.
     It is natural to call the probability of event B conditional on event A and to denote it by  P(B│A).

     Total probability formula. Bayes’s theorem

   Bayes’s theorem is a main result in probability theory, which relates the conditional and marginal probability of two aleatory events A and B. In some interpretations of probability, Bayes’s theorem explains how to update or revise beliefs in light of new evidence.

Sources

All properties of probability, the main results and theorems, including the random variables and classical probability distributions, along with suggestive examples and applications, are all exposed in a comprehensible manner in the book UNDERSTANDING AND CALCULATING THE ODDS: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life.  You may find it in the Books section with a free sample.