A Novel Approach to Probability

TL;DR

This paper introduces a new probabilistic framework that explains various natural distributions and phenomena by considering equal probability configurations, challenging traditional intuition especially in dense systems.

Contribution

It presents a novel formalism that derives known distributions like Zipf and Benford laws and explains phenomena such as photon energy quantization from a unified probabilistic perspective.

Findings

Derives Zipf and Benford laws from the formalism

Provides better predictions for election polls and market shares

Offers a probabilistic explanation for photon energy quantization

Abstract

When P indistinguishable balls are randomly distributed among L distinguishable boxes, and considering the dense system in which P much greater than L, our natural intuition tells us that the box with the average number of balls 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 in which the number of balls is smaller than the number of boxes (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 law, Benford law, particles energy distributions, and more. Further generalization of this novel approach yields not only much better predictions for election 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.