Thread: How to create stochastic matrix for these probability values?

I hope I am posting in the correct forum.

If the weather is good today, there is a 20% chance it will be fair and 7% chance it will be bad tomorrow.
If the weather is fair today, there is an 18% chance it will be good and a 22% chance it will be bad tomorrow.
If the weather is bad today, there is a 20% chance it will be good and a 30% chance it will be fair tomorrow.

(a) How to construct and label stochastic matrix describing the weather trends?

(b) What is the chance of bad weather on Thursday if on Monday there is a 88% chance of good weather and a 2% chance of bad weather.

(c) Is there a steady state vector if there is an equally likely chance of good, bad, or fair weather today?

Attempt:
As we can see
P(fair tomorrow/good today) = 0.20
P(bad tomorrow/good today) = 0.07
P(good tomorrow/fair today) = 0.18
P(bad tomorrow/fair today) = 0.22
P(good tomorrow/bad today) = 0.20
P(fair tomorrow/bad today) = 0.30

How to make the stochastic matrix? Can we do as follows?
1st column = good today
2nd column = fair today
3rd column = bad today
1st row = good tomorow
2nd row = fair tomorrow
3rd row = bad tomorrow

Therefore, the matrix will be
? ......... 0.18 ......... 0.20

.20 ...... ? ............. 0.30

.07 ...... 0.22 .......... ?

We can find the missing values assuming that the sum of entries in each column = 1
Is this correct? If no, then how to write the correct stochastic matrix?

And how to answer the other parts?
Thank you.

I figured out parts a and b. I wrote stchastic matrix (let us call it P) the way I mentioned above.
Assuming probability vector for Monday = X0, tuesday = X1 and so on, I solved part b using
X1 = P X0
X2 = P X1
X3 = P X2

But do not know how to solve part C. I first thought that I have to find a vector X such that
P X = X

But, here we do not use the information that there is an equally likely chance of good, bad, or fair weather today.