Markovian chain problems

1. A carnival man moves a pea among three shelves, A, B and C. Whenever the pea is under A, he moves it with equal probability to A or B( but not C) When it is under B, he is sure to move it to C. When the pea is under C, it is equally Likely to move it to A, B or C.

1. Set up a three state Markov chain by drawing the transition diagram and check whether it is regular

2. Find the transition matrix and evaluate the probabilities for the pea to be under each shell after two moves, if it is initially under C

3. Find the long range probabilities for the pea to be under each shell

4. Find the probabilities for the pea to be under each shell after two moves, if you know that it was initially under C and the next time it is not under B

2. A professor tries not to be late for class too often. If he is late some day, he is 90% sure to be on time the next time. If he is on time, the next day there is a 25% chance of his being late. In the long run, how often is he late for class?