A homogeneous Markov chain on {1,…,7} has transition matrix $P=\begin{pmatrix}1/3&0&2/3&0&0&0&0\\0&1/3&0&1/3&0&1/3&0\\1/2&0&1/2&0&0&0&0\\0&1/2&0&1/4&0&1/4&0\\1/2&0&0&0&1/4&1/4&0\\0&2/3&0&1/6&0&1/6&0\\1/3&1/3&0&0&0&0&1/3\end{pmatrix}$. Which state is transient?
The sets {1,3} and {2,4,6} are closed finite communicating classes, so all their states are recurrent. From 5 the chain can enter either closed class and cannot return to 5. Thus **5, D**, is transient. State 7 is also transient but is not listed.
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