  We demonstrate that the most probable state of a conserved system with a
limited number of entities or molecules is the state where non-Gaussian and
non-chi-square distributions govern. We have conducted a thought experiment
using a specific setup. We have verified the mathematical derivation of the
most probable state accurately predicts the results obtained by computer
simulations. The derived distributions approach the Gaussian and the chi-square
distributions as the number of entities approaches infinity. The derived
distributions of the most probable state will have an important role in the
fields of medical research where the number of entities in the system of
interest is limited. Especially, the non-chi-square distribution can be
interpreted by an asset distribution achieved after a repetitive game where an
arbitrary portion of one's assets is transferred to another arbitrary entity
among a number of entities with equal abilities.
