Some nice work there! I like the depth of the mathematics and the results give some reason to rethink some things.
But I would like to see the maths behind the "expected number of hits before n hits in a row" formula. I've been trying to work on something similar and I get different results: at least for n=2 and a=0.5 I get 4 hits as Expectancy value for the time until 2 hits in a row. Your formula gives 6 hits.
Either I'm doing something wrong, or you are including the 2 hits in a row in your number, or I'm taking the "expected" a bit too literally as the mathematical Expectancy Value, while you mean slightly different.
Could you give your derivation of that formula please?
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