Answer:
the proportion of days that are Sunny is 0.2
Step-by-step explanation:
Given the data in the question;
Using markov chain;
3 states; Sunny(1), Cloudy(2) and Rainy(3)
Now, based on given conditions, the transition matrix can be obtained in the following way;
![\left[\begin{array}{ccc}0&0.5&0.5\\0.25&0.5&0.25\\0.25&0.25&0.5\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%260.5%260.5%5C%5C0.25%260.5%260.25%5C%5C0.25%260.25%260.5%5Cend%7Barray%7D%5Cright%5D)
so let the proportion of sunny, cloudy and rainy days be S, C and R respectively.
such that, from column 1
S = 0.25C + 0.25R -------------let this be equation 1
from column 2
0.5C = 0.5S + 0.25R
divided through by 0.5
C = S + 0.5R ---------------------- let this be equation 2
now putting equation 2 into equation;
S = 0.25(S + 0.5R) + 0.25R
S = 0.25S + 0.125R + 0.25R
S - 0.25S = 0.375R
0.75S = 0.375R
S = 0.375R / 0.75
S = 0.5R
Therefore,
from equation 2; C = S + 0.5R
input S = 0.5R
C = 0.5R + 0.5R
C = R
Now, we know that, the sum of the three proportion should be equal to one;
so
S + C + R = 1
since C = R and S = 0.5R
we substitute
0.5R + R + R = 1
2.5R = 1
R = 1/2.5
R = 0.4
Hence, the proportion of days that are Rainy is 0.4
C = R
C = 0.4
Hence, the proportion of days that are Cloudy is 0.4
S = 0.5R
S = 0.5(0.4)
S = 0.2
Hence, the proportion of days that are Sunny is 0.2
