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Assuming I have A number of individuals who are randomly distributed into N number of groups. What is the probability of finding A (or any number a <= A) by picking M number of groups within N?

I figured that there are P(M, A) ways that the individuals can be distributed into the M groups. But I have no ideas how to move on from here.

Should it be conceived as:

Pr(Choosing M out of N | all of A are inside M)?

How should I compute the individual terms?

Jason Gay
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  • It's certainly not the conditional probability you mention. You want to compute the probability that all $A$ individuals are in the $M$ groups; this is not given. How exactly are individuals assigned to the groups? "Randomly distributed" isn't very precise. We could for example, decide on $N$ to begin with, and then assign people to the $N$ groups uniformly at random (allowing the possibility that some groups are empty, so that the original $N$ is the maximum number of groups). Or we could assign the first person to a group, and then for each succeeding person ... – saulspatz May 03 '20 at 14:36
  • ... assign him either to one of the existing groups, or a new group, uniformly at random, or we could use some other protocol. Where does the problem come from? – saulspatz May 03 '20 at 14:37

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