(Solution) - Toss a fair die repeatedly Let Sn denote the total -(2025 Original AI-Free Solution)

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Toss a fair die repeatedly. Let Sn denote the total of the outcomes through the nth toss. Show that there is a limiting value for the proportion of the first n values of Sn that are divisible by 7, and compute the value for this limit. Hint: The desired limit is an equilibrium probability vector for an appropriate seven state Markov chain.