By using Chebyshev's inequality $P(|X - E[X]| \geq \varepsilon) \leq \operatorname{Var}(X)/ \varepsilon^2$ I want to calculate the following estimation for the Catalan numbers $C_n = \frac{1}{n+1} \binom{2n}{n}$ :
$$C_n \geq \frac{4^{n-1}}{(n+1)(\sqrt{n} + \frac{1}{2})} \forall n \in \mathbb N.$$
Hint is to use a random variable $X$ which is binomial distributed, in a way that there's $\frac{1}{2}$ on the right side of Chebyshev's inequality for $\varepsilon = \sqrt{n}$, and that $\binom{2n}{n}$ is the largest binomial coefficient among $\binom{2n}{k}, k \in \{0, \ldots, 2n \}$. I got that for $X \sim \mathrm{Bin}(2n,\frac{1}{2})$. Can anybody show me how to move on?