Apologies in advance, I've not been in school for this in ages, so forgive my nomenclature.
This stack exchange discussion answers the question of the probability of a run of r successes in n trials, given that the probability of a successful trial is p.
What occurred to me is that more than one sequence of r successes can occur in a run of n. I'm able to calculate for example, the probability of
n = 20
r = 5
p = .5
which is .24987, but how can I find the probability of multiple runs of differing r?
My starting point for this is motivated by this explanation in the Upensky textbook referenced in the above stack exchange discussion here:
Following this idea, the first thing we can do is calculate the same conditions: that two runs can achieved in two mutually exclusive ways, that you've achieved both runs at the first opportunity in (r1 + r2 + 1) trials... in this case r1 and r2 are both 5
SSSSSFSSSSSS
This is the only permutation of success at the 11th trial, The odds of which are p^5(1-p)p^5. Which would be my starting point using the same "strategy" if you will, as De Moivre.
But what about more than 2 runs of variable length? e.g, given n=200 and p=.3, what is the probability of having at least 3 runs such that r1=7 r2=5 r3=9?
Any push in the right direction is greatly, greatly appreciated.
