Abstract
When P indistinguishable balls are randomly distributed among L distinguishable boxes, and considering the dense system , 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 (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.
Highlights
Probability theory is an important branch of mathematics because of its huge economic significance
The formalism of maximizing the entropy using Lagrange multipliers technique is widely used in statistical mechanics to find thermal equilibrium energy distributions i.e. [4]
An equal probability for any box yields an equal distribution of particles in the boxes
Summary
Probability theory is an important branch of mathematics because of its huge economic significance. We address the calculation of these probabilities in a unified way In statistical mechanics this kind of analysis is called maximum entropy [4], in information theory - Shannon limit [5], and in classical physics thermodynamic equilibrium [3]. The Boltzmann-Gibbs entropy is applicable to sparse systems and yields the Maxwell-Boltzmann distribution in which the boxes with average values of balls have the highest probabilities. The highest number of people have average height This is with contradistinction to the example of 12 balls distributed in 3 boxes in which the highest probability is to find an empty box, to the wealth distribution in which is it easier to find a poor man than to find a rich man. We show how various statistical laws and distributions are derived using this model
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