The Ehrenfest urn model was originally proposed as a model for diffusion of gases, but has since come to be applied in a wide variety of fields. This problem is a simplified version of the model. Consider two urns, Urn A and Urn B, containing a total of four balls. At each time interval, a ball is selected at random and transferred into the other urn. Let the states be the number of balls in Urn A. Therefore, the different states are 0, 1, 2, 3, and 4.
(a) Set up the stochastic matrix for the process.
(b) If, at some point in time, Urn A contains three balls, what is the probability that it will contain three balls after two time intervals?