On the Chvátal-Janson conjecture (Q2112271)

Chvátal conjectured that, if \(X\) is a binomial random variable with parameters \(n\) and \(m/n\), \(m\in\{0,\dots,n\}\), then for \(n\ge 2\), the probability \(q_m:=P(X\le m)\) attains its minimum when \(m\) is the integer closest to \(2n/3\). \textit{S. Janson} [Stat. Probab. Lett. 171, Article ID 109020, 10 p. (2021; Zbl 1457.60015)] established the conjecture for \(n\) large enough. In the present paper the authors prove the conjecture for every \(n\ge 2\).