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Consider the probability that a binomial random variable Bi$(n,m/n)$ withinteger expectation $m$ is at most its expectation. Chv\'atal conjectured thatfor any given $n$, this probability is smallest when $m$ is the integer closestto $2n/3$. We show that this holds when $n$ is large.

On the probability that a binomial variable is at most its expectation