What is a Markov chain?

A Markov chain is a mathematical model representing a sequence of transitions between states, where the probability of the next state depends only on the current state.

What is the Markov property?

The Markov property states that the future behavior of a system depends only on its present state, not on its past history (memorylessness).

What is a transition matrix?

A transition matrix is a square matrix containing the probabilities of transitioning between states, where the element in row $i$, column $j$ is the probability of moving from state $i$ to state $j$.

Why do rows in a transition matrix sum to 1?

Because they represent a complete set of mutually exclusive outcomes; from any state, the system must transition to one of the possible states with 100% total probability.

What is a steady-state distribution?

The steady-state distribution is a state probability vector that remains unchanged after applying the transition matrix: $\boldsymbol{\pi} P = \boldsymbol{\pi}$.

Do all Markov chains have a steady state?

Regular Markov chains (where all states communicate and are not periodic) always converge to a unique steady-state distribution.

What is an absorbing state?

An absorbing state is a state that is impossible to leave once entered; the transition probability of staying in the state is 1.

What is a stochastic matrix?

A stochastic matrix is a matrix with non-negative real entries whose rows (or columns) sum to 1, representing transition probabilities.

How do you calculate state probabilities after n steps?

Multiply the initial state vector by the transition matrix raised to the power $n$: $\mathbf{x}^{(n)} = \mathbf{x}^{(0)} P^n$.

What is a regular Markov chain?

A Markov chain is regular if some power of its transition matrix $P$ contains only positive, non-zero entries, meaning every state can eventually reach every other state.

What is a state vector?

A state vector is a probability vector representing the probability distribution of the system being in each state at a specific step.

How is Markov chain analysis used in search engines?

Google's PageRank algorithm models a user clicking links as a Markov chain, where the steady-state probability represents the popularity or importance of a web page.

math calculator

Markov Chain Transition & Steady State Calculator

Compute multi-step transition distributions and steady-state probabilities for Markov chains.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Transition Dimensions

Transition Steps n

Initial Probability Vector

State A (Init)

State B (Init)

Transition Matrix P (Rows must sum to 1.0)

A → A (p₀₀)

A → B (p₀₁)

B → A (p₁₀)

B → B (p₁₁)

Results

Steady State Vector

[66.67%, 33.33%]

State Vector after 3 steps[68.80%, 31.20%]

State Transition Diagram

Markov transition digraph showing probabilities

State Probability Convergence Table

Step	State A %	State B %
0	100.00%	0.00%
1	80.00%	20.00%
2	72.00%	28.00%
3	68.80%	31.20%

Step-by-Step Mathematical Derivations

Normalized initial state probability vector: $\mathbf{x}^{(0)} = [1.000,\; 0.000]^T$.

Normalized transition matrix rows:

\[P = \begin{pmatrix} 0.800 & 0.200 \\ 0.400 & 0.600 \end{pmatrix}\]

Multiply vector by matrix for 3 transitions:

\[\mathbf{x}^{(3)} = \mathbf{x}^{(0)} P^{3} = [0.6880,\; 0.3120]^T\]

Analytical steady-state distribution solves $\boldsymbol{\pi} P = \boldsymbol{\pi}$:

\[\boldsymbol{\pi} = [0.6667,\; 0.3333]^T\]

Formula

Markov transition & Steady State formulas

\mathbf{x}^{(n)} = \mathbf{x}^{(0)} P^n, \quad \boldsymbol{\pi} P = \boldsymbol{\pi}, \quad \sum \pi_i = 1

A stochastic matrix dictates transitions. Regular chains converge to the steady state vector independent of initial states.

Step by step

How the calculation works

1Select the matrix dimensions (2x2 or 3x3).
2Input the initial state probability distribution (A, B, C) and the transition matrix entries.
3Define the number of transition steps (n) to calculate.

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Bayes' Theorem

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math calculator

Markov Chain Transition & Steady State Calculator

Compute multi-step transition distributions and steady-state probabilities for Markov chains.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Transition Dimensions

Transition Steps n

Initial Probability Vector

State A (Init)

State B (Init)

Transition Matrix P (Rows must sum to 1.0)

A → A (p₀₀)

A → B (p₀₁)

B → A (p₁₀)

B → B (p₁₁)

Results

Steady State Vector

[66.67%, 33.33%]

State Vector after 3 steps[68.80%, 31.20%]

State Transition Diagram

Markov transition digraph showing probabilities

State Probability Convergence Table

Step	State A %	State B %
0	100.00%	0.00%
1	80.00%	20.00%
2	72.00%	28.00%
3	68.80%	31.20%

Step-by-Step Mathematical Derivations

Normalized initial state probability vector: $\mathbf{x}^{(0)} = [1.000,\; 0.000]^T$.

Normalized transition matrix rows:

\[P = \begin{pmatrix} 0.800 & 0.200 \\ 0.400 & 0.600 \end{pmatrix}\]

Multiply vector by matrix for 3 transitions:

\[\mathbf{x}^{(3)} = \mathbf{x}^{(0)} P^{3} = [0.6880,\; 0.3120]^T\]

Analytical steady-state distribution solves $\boldsymbol{\pi} P = \boldsymbol{\pi}$:

\[\boldsymbol{\pi} = [0.6667,\; 0.3333]^T\]

Formula

Markov transition & Steady State formulas

\mathbf{x}^{(n)} = \mathbf{x}^{(0)} P^n, \quad \boldsymbol{\pi} P = \boldsymbol{\pi}, \quad \sum \pi_i = 1

A stochastic matrix dictates transitions. Regular chains converge to the steady state vector independent of initial states.

Step by step

How the calculation works

1Select the matrix dimensions (2x2 or 3x3).
2Input the initial state probability distribution (A, B, C) and the transition matrix entries.
3Define the number of transition steps (n) to calculate.

Related guides

Learn more about this calculation

Guide

Standard Deviation Guide: Measure Data Spread Clearly

Understand variance, population vs sample formulas, and data spread with worked examples tied to the live DTC standard deviation calculator.

Site Guide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Learn the fastest reliable workflow for using Do The Calculation calculators, reading results, checking formulas, and using save, print, share, and export actions correctly.

Bayes' Theorem

Evaluate conditional probabilities of events.

Matrix Solver

Solve larger matrix equations and determinants.