Symmetricity of binomial distribution

Question

For each random variable, $X$, define: $Sym_X(c) = \frac{\Pr[X \geq \mu_X + c]}{\Pr[X \leq \mu_X - c]}$.

I use this definition to measure how symmetric the distribution is.

Let $X$ be a binomial random variable, $X \stackrel{}{\sim} Bin(n,p)$.

1. As a rule of thumb, I know that when $n$ is large enough, the binomial distribution is "more" symmetric
2. What is $Sym_X(c)$? (For $X \stackrel{}{\sim} Bin(n, p)$) As an expression of $n, p, c$.
3. Let $\sigma_X$ be the STD of $X$. If $c \leq \frac{1}{100}\sigma_X$ can we bound $Sym_X(c)$ by a constant which does not depend on $n,p,c$?