Consider two urns labeled A and B. There are a total of 3 orange balls and 3 blue balls. An experiment consists of picking one ball at random from each urn and interchanging the balls. This experiment is repeated independently again and again. Let Xn be the number of orange balls in urn A after n repetitions of the experiment.
1. Model {Xn, n ? 0} as a DTMC. Write down its transition matrix and draw its transition diagram.
2. Suppose that all the 3 orange balls are in urn A at the beginning, i.e., X0 = 3. Find the expected number of experiments we need to find that, for the first time, all the 3 orange balls are in urn B.
3. Suppose that all the 3 orange balls are in urn A at the beginning, i.e., X0 = 3. Find the expected number of experiments we need to find that, for the first time, all the 3 orange balls are in urn A once again.