We study a symmetric generalization $mathfrak{p}^{(N)}_k(eta, alpha)$ of the binomial distribution recently introduced by Bergeron et al, where $eta in [0,1]$ denotes the win probability, and $alpha$ is a positive parameter. This generalization is based on $q$-exponential generating functions ($e_{q^{gen}}^z equiv [1+(1-q^{gen})z]^{1/(1-q^{gen})};,e_{1}^z=e^z)$ where $q^{gen}=1+1/alpha$. The numerical calculation of the probability distribution function of the number of wins $k$, related to the number of realizations $N$, strongly approaches a discrete $q^{disc}$-Gaussian distribution, for win-loss equiprobability (i.e., $eta=1/2$) and all values of $alpha$. Asymptotic $Nto infty$ distribution is in fact a $q^{att}$-Gaussian $e_{q^{att}}^{-beta z^2}$, where $q^{att}=1-2/(alpha-2)$ and $beta=(2alpha-4)$. The behavior of the scaled quantity $k/N^gamma$ is discussed as well. For $gamma<1$, a large-deviation-like property showing a $q^{ldl}$-exponential decay is found, where $q^{ldl}=1+1/(etaalpha)$. For $eta=1/2$, $q^{ldl}$ and $q^{att}$ are related through $1/(q^{ldl}-1)+1/(q^{att}-1)=1$, $forall alpha$. For $gamma=1$, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the $Ntoinfty$ limit distribution, yielding a power law, although not exactly a $q^{LD}$-exponential decay. All $q$-statistical parameters which emerge are univocally defined by $(eta, alpha)$. Finally we discuss the analytical connection with the P{o}lya urn problem.
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