---
title: Markov Chains
description: Markov chains are mathematical models describing sequences of events where the probability of each event depends only on the state attained in the previous event. Learners will understand transition matrices, steady-state probabilities, and how to model stochastic systems.
category: mathematics
subcategory: probability-theory
difficulty: beginner, intermediate, advanced
url: /subject/markov-chains
---

# Markov Chains

Markov chains are mathematical models describing sequences of events where the probability of each event depends only on the state attained in the previous event. Learners will understand transition matrices, steady-state probabilities, and how to model stochastic systems.

## Available Resources

2 Books • 1 Courses • 2 Websites

## Books

### 1. Introduction to Probability Models

**Author:** Sheldon M. Ross

Standard applied probability textbook moving from conditional expectation to Markov chains, Poisson processes, continuous-time chains, renewal theory, queueing, reliability and simulation. Its many worked examples and exercises teach readers to build and analyse stochastic models of real systems.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0125984758?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, probability-theory

### 2. Markov Chains and Mixing Times

**Author:** David A. Levin, Yuval Peres, Elizabeth L. Wilmer

Graduate textbook on the modern theory of finite Markov chain convergence. Covers coupling, strong stationary times, spectral methods, and the cutoff phenomenon, so readers can bound how many steps a chain needs to approach its stationary distribution.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0821847392?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, probability-theory

## Courses

### 1. An Intuitive Introduction to Probability

**Author:** Karl Schmedders

University of Zurich course taught by Karl Schmedders that builds probability from everyday examples: basic rules, conditional probability, applications, discrete random variables, and the normal distribution. Learners finish able to compute and interpret probabilities in practical decisions.

**Difficulty:** Beginner | **Language:** English | **Price:** Free

**Link:** https://www.coursera.org/learn/introductiontoprobability

**Tags:** probability, conditional-probability, discrete-random-variables, normal-distribution

## Websites

### 1. probabilitycourse.com

ProbabilityCourse.com is an online learning resource that provides a structured probability course with clear explanations, worked examples, and practice problems covering topics from basics to advanced theory.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://probabilitycourse.com

**Tags:** websites, technology-computer-science, data-science--ai

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

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