ABSTRACT: We quantify the resolution with which any probability distribution may be distinguished from a displaced copy of itself in terms of a characteristic width. This width, which we call the ε resolution, is well defined for any normalizable probability distribution. We use this concept to study the broadcasting of classical probability distributions. Ideal classical broadcasting creates two (or more) output random variables each of which has the same distribution as the input random variable. We show that the universal broadcasting of probability distributions may be achieved with arbitrarily high fidelities for any finite ε resolution. By restricting probability distributions to any finite ε resolution we have therefore shown that the classical limit of quantum broadcasting is consistent with the actual classical case.