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

TL;DR

This paper establishes the tight lower bound of 1/4 for the probability that a binomial random variable exceeds its expectation when the success probability is within a specific range, refining understanding of binomial tail behavior.

Contribution

It proves the optimal constant lower bound for the probability of a binomial variable exceeding its expectation under certain parameters, improving previous bounds.

Findings

Proves that P(X > E[X]) ≥ 1/4 for p in [c/n, 1)

Identifies c as the best possible constant factor

Refines the understanding of binomial tail probabilities

Abstract

Let $X$ be a random variable distributed according to the binomial distribution with parameters $n$ and $p$. It is shown that $P(X>EX)\ge1/4$ if $1>p\ge c/n$, where $c:=\ln(4/3)$, the best possible constant factor.