I am working on solving the following problem (I am new to Markov Chains):
-It can be either rainy or sunny on a given day.
-The probability that a rainy day is followed by a rainy day is 0.5.
-The probability that a sunny day is followed by a sunny day is 0.3.
What is the percentage of days we can expect it to be rainy, asymptotically?
I created the transition matrix:
Rainy today|Sunny today
Rainy tomorrow 0.5 | 0.7
Sunny tomorrow 0.5 | 0.3
I used the joint probability formula like this P(rainy)=0.5*0.7 but this answer does not seem correct.
I hope someone can guide me and tell me what I am not doing correctly here. Thanks.