  A recent study of coherent noise model for the system size independent case
provides an exact relation between the exponent $\tau$ of avalanche size
distribution and the $q$ value of appropriate $q$-Gaussian that fits the return
distribution of the model. This relation is applied to Ehrenfest's historical
dog-flea model by treating the fluctuations around the thermal equilibrium as
avalanches. We provide a clear numerical evidence that the relation between the
exponent $\tau$ of fluctuation length distribution and the $q$ value of
appropriate $q$-Gaussian obeys this exact relation when the system size is
large enough. This allows us to determine the value of $q$-parameter \emph{a
priori} from one of the well known exponents of such dynamical systems.
Furthermore, it is shown that the return distribution in dog-flea model
gradually approaches to $q$-Gaussian as the system size increases and this
tendency can be analyzed by a well defined analytical expression.
