Suppose we have two boxes, labeled 1,...,d balls Initially some of these balls are in box 1 and the remainder are in box 2. An integer is selected at random from {1,...,d}, and the ball labeled by that integer is removed from its box and placed in the opposite box. This procedure is repeated indefinitely with the selections being independent from trial to trial. Let Xn denote the number of balls in box 1 after the n-th trial. Then Xn; n > 0, is a Markov chain on {1,...,d}.
Find the transition probabilities of this chain.
Is this chain reversible ? Justify your answer.
Calculate the stationary distribution for this chain.
For the transition matrix, i've considered cases such that first ball selected is from box 1 and such that first ball is from box 2. but i do not know how to form the transition matrix. any hints/solutions?