How many measurements (10 million molecules each time) are needed to analyze 15 million unique molecues in a 20 million pool? This is my homework: There are 20 million DNA molecules in a library for high throughput DNA sequencing, each sequencing run can generate 10 million reads (i.e., analyze 10 millions of DNA molecules), and there is variation in the selection probabilities across each DNA molecule, which follows a negative binomial distribution (p=0.5, r=0.5). The question is how many sequencing run are required to have 15 million unique DNA molecules analyzed? Can anyone help me out? Thanks a lot!
 A: Taken to an answer because it won't fit in comments. I believe if we resolve this it might be possible to begin to identify more clearly the problem you need to be solved.
I'm afraid telling me "people believe that selection probabilities in this kind of experiment follows a negative binomial distribution" doesn't address the issue I explained, so let me be clearer - such a belief is ludicrous on its face. 
See here. A negative binomial random variable takes values like 2 or 150 and doesn't take any values in $(0, 1)\,$. Probabilities take values like 0.5 or 0.357461... and don't take any values outside of $[0,1]$.
You must intend something other than the apparent meaning of your words. 
Repeating something that cannot even be approximately true doesn't resolve that fundamental mismatch. Even claiming that people believe it to be true is no use. Some people might also believe that oranges are a meter across and colored bright blue, but it's no use if you're trying to find actual oranges which simply aren't either of those things, beliefs to the contrary notwithstanding.
You must deal with the substantive issue I'm actually raising in some way. If you can't explain that mismatch, could you at least point to a document - say a paper, preferably public access - where someone else is actually claiming that probabilities are approximately negative binomial? We might be able to identify the source of confusion, either yours or theirs.
