Consider an Ehrenfest chain with 5 balls (see Example XXXXXXXXXXIf the probability mass function of X 0 , the initial number of balls in urn I, is given by find the probability that, after 6...

Consider an Ehrenfest chain with 5 balls (see Example 12.15). If the probability mass function of X₀, the initial number of balls in urn I, is given by

find the probability that, after 6 transitions, urn I has 4 balls.

Example 12.15

Suppose that there are N balls numbered 1, 2, ... , N distributed among two urns randomly. At time n, a number is selected at random from the set {1, 2, . . . , N}. Then the ball with that number is found in one of the two urns and is moved to the other urn. Let X_n
denote the number of balls in urn I after n transfers. It should be clear that {X_n
: n = 0, 1,...} is a Markov chain with state space {0, 1, 2, . . . , N} and transition probability matrix

This chain was introduced by the physicists Paul and T. Ehrenfest in 1907 to explain some paradoxes in connection with the study of thermodynamics on the basis of kinetic theory. It is noted that Einstein had said of Paul Ehrenfest (1880–1933) that Ehrenfest was the best physics teacher he had ever known.