Long Term Probability

[Asx4Life]

How do you work out long term probability without using CAS?
Insight 2011 Question 8b confused me, it said

Consider the transition matrix [0.7  0.1]  and look at the diagonal  [0.7  0.1]
                                                [0.3  0.9]                                           [0.3  0.9]

Pr(snow)=0.1/(0.1+0.3) = 1/4

WTF? Can you just do that?
Is there any other way to do this?

[aznxD]

Yes you can just do that. As that equation represents the steady state probability of that matrix.

Long run means that it's after a large number of transitions.
Ie. Sn = T^n * So , n-->infinity
So the other way to do it is let n equal and big number. 100 is suffice.
So: S100 = T^100 * So
You take the top number of the answer (which should be 0.25), as the top number of the matrix equation represents the probability of snow.
But this method can only be used in Exam 2.

[abeybaby]

for a transition matrix:
[1-a    b ]
[  a   1-b]
steady state probability for the top line is b/(a+b) and for the bottom its a/(a+b)

so if your matrix was:

Pr(snow|snow)      Pr(snow|snow')
Pr(snow'|snow)     Pr(snow'|snow')

Then your first post is correct

[Asx4Life]

I still don't get it. What do you mean by for the top line b/(a+b) and bottom line a/(a+b). I thought a steady state matrix was [1,0]

[aznxD]

Something like [1,0] is the "initial state".
Steady state is completely different, it is used to find the probability of something in the long run.
For your example:
[0.7   0.1]
[0.3   0.9]
To find the probability of snow in the long run, you use the number in the top right divided by the product of the diagonals. This is because the top line represents that it does snow.
Pr(snow)=0.1/(0.1+0.3) = 1/4
To find the probability of it not snowing, you use the number in the bottom left divided by the product of the diagonals. This is because the bottom line represents that it doesn't snow.
Pr(Snow')=0.3/(0.1+0.3) = 3/4

[Asx4Life]

Something like [1,0] is the "initial state".
Steady state is completely different, it is used to find the probability of something in the long run.
For your example:
[0.7   0.1]
[0.3   0.9]
To find the probability of snow in the long run, you use the number in the top right divided by the product of the diagonals. This is because the top line represents that it does snow.
Pr(snow)=0.1/(0.1+0.3) = 1/4
To find the probability of it not snowing, you use the number in the bottom left divided by the product of the diagonals. This is because the bottom line represents that it doesn't snow.
Pr(Snow')=0.3/(0.1+0.3) = 3/4

AHHHHH, I get it now. I was confusing steady state with initial state... Arghhhhh I'm so dumb lol. THANKS!!!!!