Markov Chain Multi-Step Transition Calculator

Calculate n-step transition probabilities using Chapman-Kolmogorov equations: P⁽ⁿ⁾ = Pⁿ

Matrix Powers

Chapman-Kolmogorov

Multi-Step Analysis

Markov Chain Configuration

Set up your Markov chain parameters and transition matrix

Number of States

One-Step Transition Matrix P

P0:

P1:

Number of Steps (n)

From State

To State

About Multi-Step Transitions

The fundamental equations for computing multi-step transition probabilities:

p_{ij}^{(u+v)} = \sum_{k \in I} p_{ik}^{(u)} p_{kj}^{(v)}

In matrix form: $P^{(u+v)} = P^{(u)} \cdot P^{(v)}$

The n-step transition matrix is simply the n-th power of the one-step matrix:

P^{(n)} = P^n = \underbrace{P \times P \times \cdots \times P}_{n \text{ times}}