Let “N” be the number of successes in a row that are happening entirely within the space of Bernoulli Trials. (Let “p” be p(success), and “q” be p(failure), where {q=1-p}). Let E(B) be the expected/mean/average number of trials needed for a streak of “N” successes in a row.
Generalize a solution of E(B) for any non-negative “N” and “p”. Generalize a solution for the variance and standard deviation of E(B). If some limits or approximations are needed, the important chunk of data is where p and N are appropriately sized with respect to each other, so E(B) does not go beyond around 10^8.
In practice, I want to get to the point where I can fill out E(B) and the STD of E(B), for any “p” and “N”:image
The numbers are fillers for now, but that is what the finished product would look like.
(Note: There may be specific assumptions or exceptions I missed that would change the nature of the problem in an incorrect direction that I don’t want to include or venture towards.)
For instance, if p=0.5 we have a coinflip, and something like THHHHHT appears during our flipping, with 5 successful heads in a row, N would be 5. I'm looking for the expected number of coin flips (trials) to produce 5 successful heads in a row, I guess this would be E(B) or Expected number of Bernoulli Trials for N successes in a row, is what I was going for. And then the standard deviation of this expected number of Bernoulli Trials for N successes in a row. (And for any p and any N)
You can find lot of works on the subject if you search on line about "consecutive k-out-of-n:F" systems. These are the systems which fail if k ( or more) consecutive components fail out of a total of n.
For instance you can read this paper,
"On success runs of a fixed length in Bernoulli sequences: Exact and asymptotic results"- Frosso S. Makri, Zaharias M. Psillakis and the other works referenced there