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Linear AlgebraChallengeMarkov Chains & Long-Run Behavior

Markov Chains & Long-Run Behavior · Challenge problem

Three cards lie in a row in their original order. Each step, with probability 1/41/4 nothing happens, and otherwise one of the three pairs of positions is chosen (each with probability 1/41/4) and those two cards are swapped. What is the probability that after exactly 44 steps the cards are back in their original order? Express your answer as a common fraction.