What fraction of days are sunny in the long run?

Question

In a certain town the weather follows a simple rule from one day to the next:

If today is sunny, tomorrow is sunny with probability $0.8$ (and rainy with probability $0.2$ ).
If today is rainy, tomorrow is rainy with probability $0.6$ (and sunny with probability $0.4$ ).

Over many years, what fraction of days are sunny?

Accepted Answer

We want the stationary distribution, the long-run fractions (\piS, \piR) that stay unchanged day to day. In steady state, the flow of probability out of "sunny" must balance the flow into it. The rate leaving sunny is \piS 	imes 0.2 (sunny days that turn rainy). The rate entering sunny is \piR 	imes 0.4 (rainy days that turn sunny). Balance: \piS \cdot 0.2 = \piR \cdot 0.4 \implies \piS = 2\,\piR. Since \piS + \piR = 1, substitute \piS = 2\piR: 2\piR + \piR = 1 \implies \piR = \frac13, \qquad \piS = \frac23. So in the long run, two-thirds of days are sunny and one-third rainy. A useful shor…

What fraction of days are sunny in the long run?

Your answer

More Probability questions