Chapter 10.2, Problem 30E

Consider the Bernoulli-Laplace diffusion model studied in Section 10.1, Exercise 30.

1. a. Let k = 5 and show that the transition matrix for the Markov chain that models the number of type I molecules in uni A is regular.
2. b. Let k = 5. In what state will this chain spend the most steps, and what fraction of the steps will the chain spend  at this state?

30. Another model for diffusion is called the Bernoulli-I.aplace model. Two urns (urn A and urn B) contain a total of 2k molecules. In this case, k of the molecules are of one type (called type I molecules) and k arc of another type (type II molecules). In addition, k molecules must be in each urn at all times. At each time step, a pair of molecules is selected, one from urn A and one from urn B. and these molecules change urns. Let the Markov chain model the number of type I molecules in urn A (which is also the number of type II molecules in urn B).

1. a. Suppose that there are j type I molecules in urn A with 0 < j < k. Explain why the probability of a transition to j − 1 type I molecules in urn A is (j/k)², and why the probability of a transition to j + 1 type I molecules in urn A is ((k − j)/k)².
2. b. Let k = 5. Use the result in pan (a) to set up the transition matrix for the Markov chain which models the number of type I molecules in urn A.
3. c. Let k = 5 and begin with all type I molecules in urn A. What is the distribution of type I molecules after 3 time steps?