  We derive simple but nearly tight upper and lower bounds for the binomial
lower tail probability (with straightforward generalization to the upper tail
probability) that apply to the whole parameter regime. These bounds are easy to
compute and are tight within a constant factor of $89/44$. Moreover, they are
asymptotically tight in the regimes of large deviation and moderate deviation.
By virtue of a surprising connection with Ramanujan's equation, we also provide
strong evidences suggesting that the lower bound is tight within a factor of
$1.26434$. It may even be regarded as the natural lower bound, given its
simplicity and appealing properties. Our bounds significantly outperform the
familiar Chernoff bound and reverse Chernoff bounds known in the literature and
may find applications in various research areas.
