Random set: A set is a random set (for someone, in certain circumstances) if for every element of the set the probability that it belongs to any subset of the set equals the probability of the subset in the set.
Formulawise: RandomSet(S) =def (s)(Si)(seS & Si inc S --> p(seSi)=p(Si)).
The necessary and sufficient condition to make a set into a random set - for someone, in certain circumstances - is that one does not know anything in particular about any of the elements of the set.
One useful way to produce a random set R for a set S is to make names on slips of paper of the same size for all the elements in S; put these slips in a vase, bowl or urn; thoroughly shake it; and blindly select from it. This is how one samples from a population.