  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.
