Consider the variation of the Ehrenfest urn model of diffusion studied in Section 10.1, Exercise 33, where one of the 2k molecules is chosen at random and is then moved between the urns with a fixed probability p.
a. Let k = 3 and suppose that p = 1/2. Show that the transition matrix for the Markov chain that models the number] of molecules in urn A is regular.
b. Let k = 3 and suppose that p = 1/2. In what state will this chain spend the most steps, and what fraction of the steps will the chain spend at this state?
c. Does the answer to part (b) change if a different value of p with 0 < p < 1 is used?

Data From Section 10.1 Exercise 33

Another version of the Ehrenfest model for diffusion starts with k molecules of gas in each urn. One of the 2k molecules is picked at random just as in the Ehrenfest model in the text. The chosen molecule is then moved to the other urn with a fixed probability p and is placed back in its urn with probability 1– p.