# Difference between revisions of "Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 30/Solution 1"

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Let us examine what happens to the individual states  Examining , we can see the 2nd column of  is  so the second state is stationary ( for all ). Likewise, .

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