Let $X_1,\cdots,X_n$ be identically distributed nonnegative, real-valued random variables that are not necessarily independent. Note that we trivially have for any $\varepsilon > 0$:
$$ \mathbb{P}\left\{ \frac{1}{n} \sum_{i=1}^{n} X_i \geq \varepsilon \right\} \leq \sum_{i=1}^{n} \mathbb{P}\{X_i \geq \varepsilon\} = n \mathbb{P}\{X_1 \geq \varepsilon\}.$$
For $n \geq 3$, is there always an example when the above inequality holds with equality, or can we obtain a sharper bound for general $n$? I am especially interested in the setting where $n$ is large.
For $n = 2$, the example $X_1, X_2 \sim U[0,1]$ with $X_2 = 1 - X_1$ and $\varepsilon = 0.5$ presents an instance where the inequality holds with an equality, but it seems hard to generalize to $n > 2$.