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Consider the Markov chain with the state space {1,2,3} and transition matrix P= .2 .4 .4...

Consider the Markov chain with the state space {1,2,3} and transition matrix

P=

.2 .4 .4
.1 .5 .4
.6 .3 .1

What is the probability in the long run that the chain is in state 1?

Solve this problem two different ways:

1) by raising the matrix to a higher power; and

2) by directly computing the invariant probability vector as a left eigenvector.

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