huh I would not have intuited that actually
Set up a state vector and a transition matrix and solve for the steady state boom
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well, the steady state is uniform, which is why the infinite-string average is uniform
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ok, i've cracked it: if you ask the question "what is the expected number of occurences of the needle in a random n-length string," you get the same number for all needles. when you ask "how many n-length strings contain the needle at least once," that's when they're different
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