  Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$.
Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over
$\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when
$m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this note we
consider the infimum value of the probability $P(X\leq E[X])$, where $X$ is a
negative binomial random variable. As a consequence, we give an affirmative
answer to the conjecture posed in [Statistics and Probability Letters, 200
(2023) 109871].
