Best lower bound on the probability of a binomial exceeding its expectation (Q2244539)

Let \(X\sim Bin\, (n,p)\), and \(c = \ln(4/3)\). In this article, the author proves that \[\mbox{ if} \quad \frac{c}{n}\leq p < 1, \quad \mbox{ then } \quad P\left(X > E[X]\right) \geq \frac{1}{4}. \] The value of \(c\) is optimal. This result is a slight improvement of a result of \textit{S. Greenberg} and \textit{M. Mohri} [Stat. Probab. Lett. 86, 91--98 (2014; Zbl 1293.60024)]. ``The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in learning theory and generalization bounds for unbounded loss functions.''