Steady State Probability Calculator

Steady-state probabilities describe long-run behavior in Markov chains and stochastic systems.

Use the result as a learning aid. For classwork, still show the formula and intermediate reasoning so the final answer is not just a black-box number.

For a Markov chain with transition matrix P, a steady-state row vector π satisfies πP = π. Its entries are long-run state proportions when the chain converges. Starting from an initial probability row vector v₀, repeated multiplication produces v₁ = v₀P, v₂ = v₁P, and so on. This calculator performs that iteration a chosen number of times.

A valid transition matrix normally has nonnegative entries and each row sums to 1. Each row describes the probabilities of moving from one current state to all possible next states. A probability state vector should also have nonnegative entries totaling 1. Under common conditions such as irreducibility and aperiodicity in a finite chain, repeated multiplication converges to a unique stationary distribution regardless of the starting distribution.

Not every chain has one attracting steady state. An identity matrix, for example, leaves every initial distribution unchanged, while periodic chains can oscillate rather than settle. Iteration should therefore be interpreted together with the structure of P.

Algebraically, a stationary distribution is a left eigenvector of P associated with eigenvalue 1, normalized so its entries sum to 1. Iteration is an intuitive numerical route; solving πP=π together with Σπᵢ=1 is the corresponding equation-based route.