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Summary It is impossible to choose at random a probability distribution on a countably infinite set in a manner invariant under permutations of that set. However, approximations to such a choice can be made by considering exchangeable probability measures on the class of probability distributions over a finite set, and letting the size of that set increase without limit. Under suitable conditions the resulting probabilities, when arranged in descending order, have non-degenerate limiting distributions. These apparently arcane considerations lead to rather concrete conclusions in certain problems in applied probability.
J. F. C. Kingmán (Mon,) studied this question.
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