Let us examine what happens to the individual states
, we can see the 2nd column of
so the second state is stationary (
Examining , we can see that every iteration this state has probability of becoming third state and of remaining the same (by the first column of ). Since is stationary, this means that
, so as
goes to infinity, the probability of
becoming the third state goes to one.
Similarly, as the number of iterations increases, the probability of becoming the second state goes to one.
Putting this all together, we have that the probability of ending up in the second state as is the sum of the probabilities of starting in the second and fourth states, i.e., , while by the same reasoning the probability of ending up in the third state is (the remaining states have probability zero). This means our answer is