I have a series of N Bernoulli tests (p, 1-p).
I need to calculate a probability of passing more than N/2 tests, depending on N and p.
The obvious solution is Chernoff bound: $\varepsilon \leq 2^{-N(p-\frac{1}{2})^2}$, but this is not sufficient for me. I actually need some stronger dependency on p. Is there any available?
I tried fitting Hoeffding, Azuma and Bernstein's inequalities, but it looks like all of these also do not give any sufficient dependency on p.
Is there any convenient estimation?
What I need is something like: $\varepsilon \leq 2^{-N*p}$