Prove Theorem 3: We fix $p, q \in [0,1]$. Let $n \geq 1$ be an integer and let $S_n$ be a sum of $n$ random variables taking values in $\{0,1\}$ mutually independent Bernoulli random variables with parameter $p$. Then
$$\mathbb{P}\left(\left|\frac{S_n}{n} - q\right| \leq \left|\frac{S_n}{n} - p\right|\right) \leq e^{-n\frac{(p-q)^2}{2}}.$$