Suppose you have 52 cards with the letters A, B, C, ..., Z and a, b, c, ..., z written on them. Consider the following Markov chain on the set S of all permutations of the 52 cards. Start with any...

Extracted text: Suppose you have 52 cards with the letters A, B, C, ..., Z and a, b, c, ..., z written on them. Consider the following Markov chain on the set S of all permutations of the 52 cards. Start with any fixed arrangement of the cards and at each step, choose a letter at random and interchange the cards with the corresponding capital and small letters. For example if the letter "M/m" is chosen, then the cards "M" and "m" are interchanged. This process is repeated again and again. (a) Markov chain. Count, with justification, the number of communicating classes of this (b) Give a stationary distribution for the chain. (c) Is the stationary distribution unique? Justify your answer. (d) initial state. Find the expected number of steps for the Markov chain to return to its