Three State Markov Chain Calculator
Propagate a three-state probability distribution through a fixed transition matrix.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Treat probability distributions as row vectors and repeatedly multiply by the same row-stochastic matrix.
p(t+1)ⱼ=Σᵢ p(t)ᵢ Pᵢⱼ; p(t)=p(0)Pᵗ
Worked example
Starting entirely in state 1 with first transition row 0.8,0.1,0.1 gives probabilities 80%,10%,10% after one step. An identity matrix preserves the initial distribution.
Assumptions and input details
Exactly three states, 0–500 steps and probabilities between zero and one. Initial probabilities and every matrix row must sum to one within 1e−10; accepted rows are normalized only for floating-point residuals. Probabilities are time-homogeneous and depend only on the current state. The table samples long histories; all requested steps are computed. No fitted transitions, individual path, stationary distribution or guaranteed convergence is inferred.
References: Cambridge: transition matrices and row distributions
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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