Markov Chain N-Step Transition Calculator

About this calculator

A Markov chain moves between states with fixed transition probabilities that depend only on the current state, and its 2×2 transition matrix P lists the chance of moving from each state to each other state in one step. Repeatedly multiplying the matrix by itself — P^n = P × P × … × P (n times) — gives the probability of ending up in each state after exactly n steps starting from any given state.

As n grows large, P^n typically settles down to rows that all look the same: the chain’s steady-state (stationary) distribution, the long-run proportion of time spent in each state regardless of where it started. This calculator computes both. Statisticians, data scientists and engineers use n-step transition math like this to model weather patterns, customer-loyalty or churn transitions, genetic inheritance (Markov models), queueing systems, and algorithms like Google’s original PageRank.