Answer:
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]](https://img.qammunity.org/2022/formulas/mathematics/college/28oiifq8w8e27j6paag9l2jtsxdmo2ky31.png)
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