**The multinomial scheme** extends the Bernoulli trial in cases with more than two outcomes. A multinomial scheme refers to a situation where you have multiple categories or outcomes and are interested in studying the probabilities of each outcome occurring.  A probability distribution that models the number of successes in a fixed number of independent trials with multiple categories is called **multinomial distribution**.

## Example
A company is surveying to gather feedback from its customers.   
The survey has three possible responses: "Satisfied," "Neutral," and "Dissatisfied." The company randomly selects `50` customers and records their responses.  
Assume that each customer is satisfied with a probability `0.3`, neutral with a probability `0.4`, and dissatisfied with a probability `0.3`.   
Calculate the probability that there will be `25` "Satisfied" responses, `15` "Neutral," and `10` "Dissatisfied".   


To solve this task multinomial distribution is used: 

import numpy as np
from scipy.stats import multinomial

# Define the probabilities of each response category
probabilities = [0.3, 0.4, 0.3]  # Satisfied, Neutral, Dissatisfied

# Specify the number of responses for which we calculate probability
response = [25, 15, 10]  # 25 satisfied, 15 neutral, 10 dissatisfied responses out of 50 total responses

# Calculate the probability mass function (pmf) using multinomial distribution
pmf = multinomial.pmf(response, n=50, p=probabilities)
print(f'Probability of {response}: {pmf:.4f}')

In the code above, we used `.pmf()` method of `scipy.stats.multinomial` class with parameters `n`  (number of trials) and `p` (probabilities of each outcome) to calculate probability that we will have certain `response` (the first argument of the `.pmf()` method. 

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

Multinomial Distribution

Example