Markov Chain Probability Calculator

Model transitions, powers, and long-run distributions confidently. Enter states, matrix values, and starting probabilities easily. Clear outputs support learning, planning, testing, and decision analysis.

Analyze transition matrices, compute n-step probabilities, inspect long-run behavior, and export results for coursework, forecasting, reliability studies, and scenario planning.

Calculator Input

Used only if initial distribution is left blank.
Use commas or new lines.
Example: 1, 0, 0 or 0.5, 0.3, 0.2
Enter one row per line. Separate values using spaces or commas.

Example Data Table

Scenario States Initial Distribution Steps Transition Matrix
Weather transition example Sunny, Cloudy, Rainy [1, 0, 0] 4 [[0.70, 0.20, 0.10], [0.30, 0.40, 0.30], [0.20, 0.30, 0.50]]
Customer loyalty model New, Active, Churned [0.30, 0.60, 0.10] 6 [[0.50, 0.40, 0.10], [0.10, 0.75, 0.15], [0.05, 0.15, 0.80]]
Machine health states Normal, Warning, Failed [0.80, 0.15, 0.05] 3 [[0.82, 0.15, 0.03], [0.20, 0.60, 0.20], [0.05, 0.10, 0.85]]

Formula Used

A Markov chain tracks how a system moves between states. If the current state probabilities are stored in row vector v and the transition matrix is P, then the distribution after n steps is:

v(n) = v(0) × Pn

Each row of P must sum to 1 because that row describes every possible next-state outcome from one current state. The entry in row i and column j is the probability of moving from state i to state j in one step.

The long-run or steady-state vector π satisfies:

π = π × P and sum(π) = 1

This calculator estimates the steady-state vector with repeated multiplication until changes become negligible.

How to Use This Calculator

  1. Enter the number of states in your system.
  2. Add state labels using commas or separate lines.
  3. Paste the transition matrix with one row per state.
  4. Enter an initial distribution, or leave it blank to use the selected start state.
  5. Set the number of steps and choose the target state.
  6. Click Calculate Probabilities to generate the results section above the form.
  7. Use the export buttons to download CSV or PDF output for reporting.

Frequently Asked Questions

1. What does this calculator measure?

It computes future state probabilities in a discrete Markov chain, including transition matrix powers, target-state probability, and an estimated steady-state distribution.

2. Why must each row sum to 1?

Each row lists all next-step outcomes from a current state. Since one of those outcomes must occur, their probabilities must total exactly 1.

3. What is Pn in the output?

Pn is the n-step transition matrix. It shows the probability of moving from any starting state to any ending state after n transitions.

4. Can I leave the initial distribution blank?

Yes. If left blank, the calculator creates a one-hot starting vector using the chosen start state index.

5. What does steady-state mean?

Steady-state is the long-run distribution reached when repeated transitions stop changing the probability pattern meaningfully. Some chains converge quickly; others may not.

6. How many states can I enter?

This version accepts 2 to 8 states. That range keeps the page readable while still supporting many classroom and business scenarios.

7. When should I use this tool?

Use it for weather models, customer retention, machine reliability, inventory movement, queue behavior, and any process where next outcomes depend only on the current state.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.