Probability and Thermodynamics

When P indistinguishable balls are randomly distributed among L distinguishable boxes, and considering the dense system P >> L, our natural intuition tells us that the box with the average number of balls P/L has the highest probability and that none of boxes are empty; however in reality, the probability of the empty box is always the highest. This fact is with contradistinction to sparse system P << L (i.e. energy distribution in gas) in which the average value has the highest probability. Here we show that when we postulate the requirement that all possible configurations of balls in the boxes have equal probabilities, a realistic "long tail" distribution is obtained. This formalism, when applied for sparse systems, converges to distributions in which the average is preferred. We calculate some of the distributions resulted from this postulate and obtain most of the known distributions in nature, namely: Zipf’s law, Benford’s law, particles energy distributions, and more. Further generalization of this novel approach yields not only much better predictions for elections, polls, market share distribution among competing companies, and so forth, but also a compelling probabilistic explanation for Planck's famous empirical finding that the energy of a photon is hv. This paper unifies everyday probability calculations done by surveyors, gamblers and economists with the calculations done by physicists, therefore enabling the applications of the methodology of statistical physics in economics and the life sciences.