# A Markov chain has the transition probability: P = 0 .42 .58 0 1 0 .54 0 .46 Find `P^(2)`

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Given the transition matrix `P=([0,.42,.58],[0,1,0],[.54,0,.46])` we are asked to find `P^2` . (This is the matrix after 2 repetitions of the experiment.)

`P^2=P*P=([.3132,.42,.2668],[0,1,0],[.2404,.2268,.5248])`

(Just use matrix multiplication or technology.)

The entries in this matrix give the probabilities of transitioning from one state to another after 2 repetitions of the experiment.

Thus the probability of transitioning from state 3 to state 2 originally was 0, but after another repetition it is .2268.