  In a recent paper, Svante Janson has considered a conjecture suggested by
Va\v{s}ek Chv\a'atal dealing with the probability that a binomial random
variable with parameters $n$ and $m/n$ - where $m$ is an integer - exceeds its
expectation $m$. Albeit Janson has provided a proof of this conjecture for
large $n$, we show that the result actually holds for each $n\geq 2$.
