Another model for diffusion is called the Bernoulli-Laplace 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 are 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).
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.
b. Let k = 5. Use the result in part (a) to set up the transition matrix for the Markov chain that models the number of type I molecules in urn A.
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?