Answer:
F=y+z=4/6.25
Step-by-step explanation:
First, we have to consider that in the problem model we have only two possible states: sunny and cloudy. Now, according to the information given in the statement, we also have the behavior of the last two days. In any case, we can have four possible transitional states:
Today-Yesterday(S=Sunny, C=cloudy)
1) ST and SY (Sunny today and sunny yesterday)
2) ST and CY.
3) CT and SY.
4) CT and CY.
Now, according to the statement, the probabilities given for the four states can be expressed by the following matrix:
Now, making w, x, y, z as the transition probabilities for the four states mentioned, we then have that:
x=0.6w+0.5x
w=1.25x (1)
x=0.4y+0.2z (2)
y=0.4w+0.5x
y= 0.4(1.25x)+0.5x=x
y=x (3)
replacing 3 in 2:
y=0.4y+0.2x
x=3y (4)
And as w+x+y+z= 1 (no more possible combinations):
w+x+y+z=1 (5)
So, replacing the expressions obtained previously in equation 5, we have finally that:
1.25x+x+x+3x=1
x=1/6.25=y
z=3x=3/6.25
So, the fraction of sunny days is given by:
F=y+z=4/6.25
