Markov Chain Calculator

Work with a Markov chain: n-step transition probabilities, the distribution after n steps, the stationary distribution and absorbing states, with exact fractions.

Transition matrix P (rows add to 1)	0.7 0.3 0.4 0.6
Starting distribution (optional)	1 0
Number of steps n	3

Transition matrix P (rows add to 1)

Row i gives the probabilities of moving from state i to each state. Fractions like 1/3 are fine.

Starting distribution (optional)

One probability per state, adding to 1.

Number of steps n

After 3 steps

(583/1000, 417/1000)

In the long run the chain spends 57.1429% of the time in state 1, 42.8571% of the time in state 2 (if it is regular, Pⁿ tends to a matrix with π in every row).

Exact fraction. Checked exactly.

Method: Matrix powers and the stationary equations
Worked out with: AllCalcHub exact fractions
Check: Every step that can be checked was checked; πP = π exactly

Distribution after 3 steps: (583/1000, 417/1000)
Stationary distribution π: (4/7, 3/7)

P^3

	1	2
1	583/1000	417/1000
2	139/250	111/250

About the Markov Chain Calculator

A Markov chain moves between states with fixed probabilities that depend only on the current state. The n-step transition probabilities are the entries of Pⁿ, and a starting distribution p₀ becomes p₀Pⁿ after n steps.

A stationary distribution π satisfies πP = π with its entries adding to 1; for a regular chain every starting distribution converges to it.

The formula

pₙ = p₀Pⁿ; πP = π, Σπᵢ = 1

Absorbing state: Pᵢᵢ = 1: once there, the chain stays

Worked example

P = [[0.7, 0.3], [0.4, 0.6]]

πP = π: 0.3π₁ = 0.4π₂
π = (4/7, 3/7)

Questions people ask

Do I need decimals?

No: fractions keep the answer exact, and decimals like 0.7 are read exactly too.

What is a steady state?

A distribution π with πP = π: once the chain reaches it, the probabilities stop changing. A regular chain settles to the same steady state from any starting point.