In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. We have shown that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$. Let $n$ be a non-zero natural number, $p$ an element of the interval $]0,1[$ and $Z$ a random variable following a binomial distribution with parameter $(n, p)$. Using the previous question, bound $\mathbb{P}\left(\left|\frac{Z}{n}-p\right| \geqslant \varepsilon\right)$ as a function of $n, p$ and $\varepsilon$.
In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. We have shown that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$.
Let $n$ be a non-zero natural number, $p$ an element of the interval $]0,1[$ and $Z$ a random variable following a binomial distribution with parameter $(n, p)$. Using the previous question, bound $\mathbb{P}\left(\left|\frac{Z}{n}-p\right| \geqslant \varepsilon\right)$ as a function of $n, p$ and $\varepsilon$.