Solution
Sunny in the Long Run
Show the problem again
In a certain town each day is either sunny or rainy. If today is sunny, tomorrow is sunny with probability 0.8; if today is rainy, tomorrow is sunny with probability 0.4. Over a very long period, what fraction of days are sunny? Give a reduced fraction.
Worked solution
The answer is 2/3. Let π be the long-run fraction of sunny days. In equilibrium the flow from sunny to rainy equals the flow from rainy to sunny: 0.2π = 0.4(1 − π), so 0.6π = 0.4 and π = 2/3. The starting weather is irrelevant: a two-state chain with both transition probabilities strictly between 0 and 1 forgets its initial state geometrically fast.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.