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We generalise the known fact that for binomial $X_{n,k} \sim \mathrm{Bin}(n,k/n)$ one has $\inf_{k>1,n} \mathrm{P}(X_{n,k} \geq k) \geq \lim_{k \to1+}\mathrm{P}(X_{2,k} \geq k) = 1/4$ to cover probabilities of exceeding aconstant shift away from the mean. The proof is very short and the theoremyields the original result as a special case, as well as proving that ananalogous result holds for the Poisson distribution. We furthermore prove asimilar property holds for the family of beta distributions near their mode.Thanks to the connection between Binomial and Beta distributions, this allowsus to shed some further light on the original result regarding Binomialprobabilities.

Another look at binomial and related distributions exceeding values close to their centre