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Markov chains and steady states

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Useful for: IB Math AI HL

Learn it A Markov chain moves between states with fixed probabilities. Multiply the current distribution by the transition matrix to get the next one. In the long run it settles to a steady state where the distribution no longer changes.

Example: If sunny today, 80% sunny tomorrow; if rainy, 40% sunny tomorrow. Long-run fraction of sunny days?

  1. Steady state: s = 0.8s + 0.4(1 − s) → 0.6s = 0.4.
  2. s = 2/3.

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