Every week day, a broker is investing in one of two stocks. In any given week day, stock 1 goes up with probability 0.8 and goes down with probability 0.2, while stock 2 goes up with probability 0.7 and goes down with probability 0.3. If the stock that he invests in today goes up, then he will invest in stock 1 tomorrow, and if it goes down, then he will invest in stock 2 tomorrow.

a) Let X_n = i if he invests in stock i on day n, for i = 1, 2. Find the matrix of transition probabilities of the Markov chain (X_n)n1.

b) What is the probability that the stock that he will invest on Thursday goes up, given that the stock that he invested on Monday went up?

c) Suppose that initially (i.e. on day 0) he chooses to invest in stock 1 or stock 2 with equal probabilities. What is the probability that he will invest in stock 1 on day 3?