In view of the dense system P ≫ L, when P indistinguishable balls are randomly distributed among L distinguishable boxes, our natural intuition tells us that the box with the average number of P/L balls has the highest probability and that none of the boxes are empty; however, the probability of the empty box is always the highest in fact. This fact is contrary to the sparse P ≫ L method (i.e. the distribution of energy in gas) in which the average value is most probable. Here we show that a practical “long tail” distribution is obtained when we postulate the requirement that all possible configurations of balls in the boxes have equal probabilities. When applied to sparse systems, this formalism converges into distributions where the average is favored. Some of the distributions resulting from this postulate are determined and most of the established distributions in nature are obtained, namely: Zipf’s law, Benford’s law, energy distributions of particles, and more. Further generalization of this new method not only yields much better forecasts for elections, surveys, distribution of market share among competing firms, and so on, but also a convincing probabilistic explanation for Planck’s popular empirical finding that a photon’s energy is hv. This paper unifies surveyors, gamblers and economists’ regular likelihood estimates with physicists’ calculations, thus allowing for the application of statistical physics methods in economics and life sciences.
Kafri Nihul Ltd., Tel Aviv, Israel.
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