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 $0.29/n \le p < 1$ with probability at least $1/4$ strictly exceeds its expectation. We also show that for $1/n \le p < 1 - 1/n$, $X$ exceeds its expectation by more than one with probability at least $0.0370$. Both probabilities approach $1/2$ when $np$ and $n(1-p)$ tend to infinity.