1. A ?ea moves around the vertices of a triangle in the following manner: Whenever it is at vertex i it moves to its clockwise neighbor vertex with probability pi and to the counterclockwise neighbor with probability qi = 1 - pi , i = 1, 2, 3.

(a) Find the proportion of time that the ?ea is at each of the vertices.

(b) How often does the ?ea make a counterclockwise move that is then followed by ?ve consecutive clockwise moves?

2. Consider a Markov chain with states 0, 1, 2, 3, 4. Suppose P0,4 = 1; and suppose that when the chain is in state i, i > 0, the next state is equally likely to be any of the states 0, 1, ... , i - 1. Find the limiting probabilities of this Markov chain.