We have already considered that if events A and B are independent then P(A and B) = P(A) \*P(B). This formula is a special case of the more general **probabilities multiplication rule**:

It states that the probability of the joint occurrence of two events, A and B, is equal to the probability of event A multiplied by event B's conditional probability, given that event A has occurred.   
*Let's look at the example for a better understanding of this rule*:  

Assume you draw two cards from a standard deck (52 cards) without replacement. What is the probability of drawing a heart on the first card and a diamond on the second?   
Event A - drawing a heart first. 
Event B - drawing a diamond second.

import numpy as np

# Creating a deck of 52 cards
suits = ['H', 'D', 'C', 'S']  # Hearts, Diamonds, Clubs, Spades
ranks = ['2', '3', '4', '5', '6', '7', '8', '9', '10', 'J', 'Q', 'K', 'A']
deck = [rank + suit for suit in suits for rank in ranks]

# Counting the number of cards in the deck
total_cards = len(deck)

# Counting the number of hearts and diamonds in the deck
hearts = sum(card[-1] == 'H' for card in deck)
diamonds = sum(card[-1] == 'D' for card in deck)

# Calculating P(A)
p_A = hearts / total_cards

# Calculating P(B|A) 
# We have already removed one heart from the deck
# Total number of cards has become 1 less
# As a result conditional probability can be calculated
p_B_cond_A = diamonds / (total_cards - 1)

# Resulting probability due to multiplication rule
p = p_A * p_B_cond_A
print(f'Resulting probability is {p:.4f}') 

>Note
>
>Pay attention that in the multiplication probabilities rule, the order in which events occur is unimportant - we can consider both the probability *P(B)\*P(A|B)* and *P(A)\*P(B|A)*. 

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

The Multiplication Rule of Probability