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Consider a Markov chain with state space {1,2,3} and transition matrix. P= .4 .2 .4 .6...

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

P=

.4 .2 .4
.6 0 .4
.2 .5 .3

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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