---
title: "Markov Chains and the Memoryless Assumption"
description: "The next state depends on the current state and on nothing before it. That single assumption turns a process into a matrix, and questions about the far future become questions about matrix powers."
canonical: https://lightmysky.com/learn/mathematics/markov-chains-and-the-memoryless-assumption-mt_2wDa_JxJ_o
source: https://lightmysky.com/learn/mathematics/markov-chains-and-the-memoryless-assumption-mt_2wDa_JxJ_o.md
retrieved: 2026-09-12
---

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# Markov Chains and the Memoryless Assumption

The next state depends on the current state and on nothing before it. That single assumption turns a process into a matrix, and questions about the far future become questions about matrix powers.

Subject: Mathematics · Area: Probability · Ages 22 to 24
Page: https://lightmysky.com/learn/mathematics/markov-chains-and-the-memoryless-assumption-mt_2wDa_JxJ_o

## Ready when they can

- Write the transition matrix for a described process
- Compute an n-step transition probability
- Give a process that is not Markov and say what extra memory would make it one

## Lesson: Tomorrow depends only on today

A Markov chain is a process where tomorrow depends only on today, never on earlier history. That single memoryless assumption turns the process into a matrix. Rows are the current state and columns are the next state, and each row sums to 1 because tomorrow always brings some state. For a Sunny and Cloudy model, the Sunny row might read 0.8 to Sunny then 0.2 to Cloudy, while the Cloudy row reads 0.4 then 0.6.

**Example.** Start from Sunny with the rows above. The chance of Sunny two days out needs both paths: Sunny then Sunny gives 0.8 times 0.8, which is 0.64, and Sunny then Cloudy then Sunny gives 0.2 times 0.4, which is 0.08. Add the disjoint paths to get 0.72. Tree diagrams draw the same idea: multiply along each branch path, then add across paths that land together.

Matrix powers carry this further: the square of the transition matrix holds every two-step probability, the cube holds every three-step one. From Sunny, the chance of Cloudy two days out is 0.8 times 0.2 plus 0.2 times 0.6, which is 0.16 plus 0.12, giving 0.28. Every n-step question is the multiplication rule in disguise, chaining today into tomorrow and summing over middles.

**Tip.** A rule using the last two days is not Markov, since older history matters. The fix is to enlarge the state so it carries the memory: track pairs of consecutive days as the state. Given a pair like (Sunny, Cloudy), tomorrow depends only on the pair. When memory breaks the rule, a bigger state restores it.

**Recap.** One-step chances form a matrix, multi-step chances come from its powers, and memory needs a bigger state.

## Practice

13 questions on this page, each with its working shown.

## Needs first

- [Weak Solutions and Test Functions](https://lightmysky.com/learn/mathematics/weak-solutions-and-test-functions-mt_AKlC3Ol70p)
- [Conditional Expectation Given a Sigma-Algebra](https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8)
- [Conditional Probability, Independence and Bayes' Theorem](https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK)
- [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL)

## Opens up

- [Markov Decision Processes: States, Actions and Return](https://lightmysky.com/learn/computing/markov-decision-processes-states-actions-and-return-mt_bTCO27sgc6)
- [Stationary Distributions and Long-Run Behaviour](https://lightmysky.com/learn/mathematics/stationary-distributions-and-long-run-behaviour-mt_MGfHvUGkeh)