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Runs on a Ring
This is a combined math and Mathematica problem. Suppose one has a ring of n sites. Each site is populated by a zero or a one, each with random probability one half. What are the odds that in a ring of size n there is at least one run of ones (or, equivalently, zeros) of length t? The problem is solved for lists of site n. See http://mathworld.wolfram.com/Run.html for an excellent discussion, but I have not been able to find a solution for rings. I have been able to solve the problem inelegantly for runs of length 3 by using the package RSolve to solve some coupled difference equations generated largely from observation.