Simple Probability Matrix

Question

Question:

Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your upcoming test. If you fail your previous test, then you have 0.5 chance you will fail your upcoming test. If it continues over a long time, what is the probability that you will pass a test?

I have calculated the eigenvalues and the corresponding eigenvectors of P, but I don't know where to go after that. Any help would be appreciated.

Answer 1

Sounds like you're on the right track. Assuming you've set up the 2x2 matrix that represents the transitions correctly, then you're all set. Recall that the long-run distribution is interpreted as the stationary probability vector, which is the left eigenvector of the transition matrix P associated with eigenvalue 1.