  Let $S$ and $X$ be independent random variables, assuming values in the set
of non-negative integers, and suppose further that both $\mathbb{E}(S)$ and
$\mathbb{E}(X)$ are integers satisfying $\mathbb{E}(S)\ge \mathbb{E}(X)$. We
establish a sufficient condition for the tail probability $\mathbb{P}(S\ge
\mathbb{E}(S))$ to be larger than $\mathbb{P}(S+X\ge \mathbb{E}(S+X))$. We also
apply this result to sums of independent binomial and Poisson random variables.
