Question

A certain city prides itself on having sunny days. If it rains one day, there is a 90% chancethat it will be sunny the next day. If it is sunny one day, there is a 30% chance that it willrain the following day. (Assume that there are only sunny or rainy days.)

Required:
Does the city have sunny days most of the time? In the long run, what fraction of all days is sunny?

Answer 1

1/2 of all the days in the city are sunny , also the city does not have sunny days most of the time

Explanation:

Given data :

90% chance of sunny day when it rains the previous day

30% chance of rain when it is sunny the previous day

Determine :

If the city has sunny days most of the time and the fraction of the days that is sunny

First step : construct a transition matrix

`P = \left[\begin{array}{ccc}0.70&0.3&\\0.10&0.90&\\\end{array}\right]`

lets assume the frequency of the long = β = [ β1, β2 ]

from our matrix equation we can say ; β = βP

β1 = 0.70β1 + 0.30β2 ---- ( 1 )

β2 = 0.10β1 + 0.90β2 ----- ( 2 )

Given that : β1 + β2 = 1 , hence β2 = 1 - β1

plug this value into equation 1

β1 = 0.70β1 + 0.30( 1 - β1 )

∴ β1 = 0.5

also β2 = 1 - 0.5 = 0.5

( This means that half of all the days in the city are sunny ) also the city does not have sunny days most of the time