Intuitively, it makes sense that if $X\sim B(n,p)$, assuming that $np\in \mathbb{Z}$, then $P(X\leq E[X])\geq 1/2$. Wikipedia confirms this in the median section https://en.wikipedia.org/wiki/Binomial_distribution#Median.
However, the results there come from research papers, some still quite recent. Is there a more well-known way to show that $\left \lfloor{np}\right \rfloor \leq m \leq \left \lceil{np}\right \rceil$, where $m:=$ median?
I think the biggest problem comes from the binomial distribution not being symmetric.
