A certain city prides itself on having sunny days f it rains

A certain city prides itself on having sunny days. f it rains one day, there is a 90% chance that it will be sunny the next day. If it is sunny one day, there is a 30% chance that it will rain the following day. (Assume that there are only sunny or rainy days.) Does the city have sunny days most of the time? n the long run, what fraction of all days is sunny?

Solution

This is a markov chain with transition matrix
.. .. .. . sunny rainy
sunny 0.7 .... 0.3
rainy.. 0.9 ... 0.1

It is obvious that the city will have sunny days most of the time.

To figure out the exact long run probability, solve
0.7p + 0.9q = p
0.3p + 0.1q = q
p + q = 1,
which yields the steady state vector as [0.75 0.25],
ie 3/4 of the days are sunny.