Long-run share of a two-state chain

MediumLinear Algebra~4m

A machine is either Up or Down, checked each day. If it is Up today, it is Up tomorrow with probability $0.8$ (else Down). If it is Down today, it is Up tomorrow with probability $0.4$ (else Down). Over the long run, what fraction of days is the machine Up? Give a decimal to four places.

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The long-run share solves $\pi P=\pi$ with $\pi$ summing to $1$ .
Balance equation: $u=0.8u+0.4d$ , and $u+d=1$ .

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0.6667 (± 0.001)

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Long-run share of a two-state chain

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