Bound for deviation from mean of binomial random variable Let $S_n$ denote a binomial $(n,p)$ random variable. The problem is to show that there is a constant $C$ depending only on $p$ such that for all $\epsilon > 0,$
$$P(|S_n/n - p| > \epsilon) \le \frac{C}{n^{10} \epsilon^{20}}.$$
The hint is to use Markov's inequality, but I can't see how it applies... can anyone give another hint or some further advice? Thanks in advance.

You might want to use Chebyshev's inequality, which is a corollary of Markov's inequality (see: http://en.wikipedia.org/wiki/Markov%27s_inequality)
\begin{equation}
P( |X - \mathbb{E}X| \geq \varepsilon ) \leq \frac{\text{Var}(X)}{\varepsilon^2}
\end{equation}
for random variable $X$.  It's a special case of Markov's inequality found by considering the function $(X - \mathbb{E}X)^2$.  Then you can use the usual rules for $\mathbb{E}[aX]$ and Var$[aX]$ for the random variable $S_n/n$, though I think the denominator in your question might be $n\varepsilon^2$ rather than $n^{10}\varepsilon^{20}$.
