# Reducible Markov Chain with a state that communicates with nothing

Lets say there is a Markov chain whose transition matrix is defined as follows

$$P = \left( \begin{array}{cccc} 0.5 & 0.5 & 0 & 0 \\ 0 & 0.25 & 0 & 0.75 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right)$$

Now I've been told that communicating classes are those subsets of states communicate with each other, i.e. $P_{ij} > 0$ and $P_{ji} > 0$.

In the example above 1 communicates only with itself, 2 and 4 communicate with each other, but state 3 doesn't communicate with any other state, even itself.

So the communicating classes are {1} and {2,4}. But is {3} a such a class? It seems odd for it not to be, but it doesn't seem to match the definition as I've seen it. Presumably the chain is still reducible?

Edit: Fixed an issue where initially the top row of P failed to sum to one