If $X$ is a binomial random variable with parameters $n$ and $p$, then by known tail bounds:
$$ \Pr[X\leq k] \leq \exp\left(-2 \frac{(np-k)^2}{n}\right),\!$$
and if there are $m$ such i.i.d. random variables, $X_1,\dots,X_m$, then the probability that at least one variable is less than $k$ is:
$$ \Pr[X_1\leq k \cup \dots \cup X_m\leq k] \leq m\cdot\exp\left(-2 \frac{(np-k)^2}{n}\right),\!$$ by the union bound.
What happens if the variables $X_1,\dots,X_m$ are statistically dependent? Is the second inequality still true?