Answer:
0.628
Step-by-step explanation:
From the information given:
Let S represent sunny days and C represent cloudy days;
S is expressed in the matrix to be in the first row and first column
C is expressed in the matrix to be in the second row and second column
Then, the transition for the probability matrix can be computed as follows:
[tex]P = \right. \left[\begin{array}{cc}0.7&0.3\\0.5&0.5\\\end{array}\right][/tex]
[tex]P^2 = \right. \left[\begin{array}{cc}0.7&0.3\\0.5&0.5\\\end{array}\right] \right. \left[\begin{array}{cc}0.7&0.3\\0.5&0.5\\\end{array}\right][/tex]
[tex]P^2 = \right. \left[\begin{array}{cc}0.49+0.15&0.21+0.15\\0.35+0.25&0.15+0.25\\\end{array}\right] \right.[/tex]
[tex]P^2 = \right. \left[\begin{array}{cc}0.64&0.36\\0.60&0.40\\\end{array}\right] \right.[/tex]
[tex]P^3 = \right. \left[\begin{array}{cc}0.64&0.36\\0.60&0.40\\\end{array}\right] \right. \right. \left[\begin{array}{cc}0.7&0.3\\0.5&0.5\\\end{array}\right][/tex]
[tex]P^3 = \right. \left[\begin{array}{cc}0.628&0.372\\0.62&0.38\\\end{array}\right] \right.[/tex]
So, if it is sunny, the probability that it is going to be sunny 3 days later = 0.628
Using the transition of the events, it is found that there is a 0.628 = 62.8% probability that it will be sunny 3 days later.
The transition of events, considering that it is sunny today and we want it to be sunny 3 days later, is given by:
S - S - S - S
S - C - S - S
S - S - C - S
S - C - C - S
Considering the probabilities stated in the exercise(S - S: 0.7, S - C: 0.3, C - S: 0.5, C - C: 0.5), for the last three days, the probability is:
[tex]p = 0.7^3 + 0.3(0.5)(0.7) + 0.7(0.3)(0.5) + 0.3(0.5)(0.5) = 0.628[/tex]
0.628 = 62.8% probability that it will be sunny 3 days later.
A similar problem is given at https://brainly.com/question/22410644
