An Elementary Analysis of the Probability That a Binomial Random Variable Exceeds Its Expectation

We give an elementary proof of the fact that a binomial random variable $X$ with parameters $n$ and $0.29\mathrm{/}n\le p<1$ with probability at least $1\mathrm{/}4$ strictly exceeds its expectation. We also show that for $1\mathrm{/}n\le p<1-1\mathrm{/}n$, $X$ exceeds its expectation by more than one with probability at least $0.0370$. Both probabilities approach $1\mathrm{/}2$ when $np$ and $n(1-p)$ tend to infinity.