1. A box contains two balls, either of which can be red or yellow. At each stage, a ball is chosen at random and replaced with a ball of the opposite colour. Let Xn be the number of red balls in the box at time n. Find the transition matrix.
2. (Random walk with absorbing barriers) Consider a random walk on four consecutive points 1, 2, 3, 4 on a line. Starting at either 2 or 3, the process moves right with probability 2/3 or left with probability 1/3. Whenever the process gets to either 1 or 4, it remains there forever. Classify the four states as to recurrent or transitive, and for each pair (i, j) of transitive states find the expected number of times the process will be in j starting from state i.