Course Content

Probability Theory Basics

1. Basic Concepts of Probability Theory

Stochastic Experiment and Random Event Probability and It's Properties Geometrical Probability Challenge: Solving the Task Using Geometric Probability Independence and Incompatibility of Random Events Conditional Probability

2. Probability of Complex Events

Inclusion-Exclusion Principle Challenge: Solving the Task Using Inclusion-Exclusion Principle The Multiplication Rule of Probability Law of Total Probability Bayes' Theorem Challenge: Solving the Task Using Bayes' Theorem

3. Commonly Used Discrete Distributions

Binomial Distribution Challenge: Solving Task Using Binomial Distribution Multinomial Distribution Geometric Distribution Poisson Distribution Challenge: Solving Task Using Poisson Distribution

4. Commonly Used Continuous Distributions

Continuous Uniform Distribution Exponential Distribution Challenge: Solving Task Using Exponential Distribution Gaussian Distribution Challenge: Solving Task Using Gaussian Distribution

5. Covariance and Correlation

What is Covariance?What is Correlation?Challenge: Solving the Task Using Correlation

Multinomial Distribution

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:


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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.

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Section 3. Chapter 3