I'm doing DNA library screening, and I meet a difficult probability question. I'm looking for some help here in the bioinformatics part.

I mean nothing of bribe or disrespect. If you don't like the thread title, please neglect it, as I found I cannot change the title after posting.

______________ We can absolute the biological question as the following pure math question:

There are [x] balls, including [3] red balls and [x - 3] grey balls. Every 96 balls are located randomly in one bag, so there are [n = x/96] bags. Now, I want to divide those bags into [m] equal teams. What is the probability that no more than one red ball locates in one team? (x & n are constant, m is the only variable.)

(I expect the answer and the explanation. Thank you very much! Please help me~~~~~)

_________________The real case is as follows:

I have a DNA library of 161280 clones, which are located in 1680 96-well plates. The library contains ~3x coverage of our model organism's genome, so there will be ~3 positive clones for one single copy gene in this library. My work is to find the positive clones of some genes of interest.

At first, I should divide those 96-well plates into [m] teams, and mix all samples in each team in order to detect which team has the positive clone we want. Then, using 3 or higher dimension based method to detect which well has the positive clone we want from that team. Because higher dimension screening method causes more false positive results, the less probability that one team has more than one positive clone, the better!

Accordingly, the point is more teams means less false positive results, however, it also means more money, more time, et. al.

Now, I want to calculate the probability that no more than one positive clone locates on one team, after dividing those plates into [m] teams. Finally, I can decide how many teams [m] I should make, according to the probability result above as well as money, time, labor, and some other factors.

(We should perform the screening work using a company's robot before Christmas and pay a lot of money, so I really want to calculate the probability as soon as possible in order to confirm the team number [m])

I mean nothing of bribe or disrespect. If you don't like the thread title, please neglect it, as I found I cannot change the title after posting.

______________ We can absolute the biological question as the following pure math question:

There are [x] balls, including [3] red balls and [x - 3] grey balls. Every 96 balls are located randomly in one bag, so there are [n = x/96] bags. Now, I want to divide those bags into [m] equal teams. What is the probability that no more than one red ball locates in one team? (x & n are constant, m is the only variable.)

(I expect the answer and the explanation. Thank you very much! Please help me~~~~~)

_________________The real case is as follows:

I have a DNA library of 161280 clones, which are located in 1680 96-well plates. The library contains ~3x coverage of our model organism's genome, so there will be ~3 positive clones for one single copy gene in this library. My work is to find the positive clones of some genes of interest.

At first, I should divide those 96-well plates into [m] teams, and mix all samples in each team in order to detect which team has the positive clone we want. Then, using 3 or higher dimension based method to detect which well has the positive clone we want from that team. Because higher dimension screening method causes more false positive results, the less probability that one team has more than one positive clone, the better!

Accordingly, the point is more teams means less false positive results, however, it also means more money, more time, et. al.

Now, I want to calculate the probability that no more than one positive clone locates on one team, after dividing those plates into [m] teams. Finally, I can decide how many teams [m] I should make, according to the probability result above as well as money, time, labor, and some other factors.

(We should perform the screening work using a company's robot before Christmas and pay a lot of money, so I really want to calculate the probability as soon as possible in order to confirm the team number [m])

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