Is this probability inequality always true? For $n$ random variables $X_1$, $X_2$, $\dotsc$, and $X_n$.
Is it always true that:
$$\mathbb{P}\left[\sum_{k=1}^{n} X_k>a\right]\geq\mathbb{P}\left[\max\{X_1, X_2, \dotsc, X_n\}>a\right].$$
Or is it true when all the variables are i.i.d?
P.S. $\mathbb{P}\left[\mathcal{A}\right]$ denotes the probability of event $\mathcal{A}$
 A: If they are positive:
$$\sum_{k=1}^{n} X_k\leq a\Rightarrow\max\{X_1, X_2, \dotsc, X_n\}\leq a$$
Then:
$$\mathbb{P}\left[\sum_{k=1}^{n} X_k\leq a\right]\leq\mathbb{P}\left[\max\{X_1, X_2, \dotsc, X_n\}\leq a\right]$$
Then:
$$1-\mathbb{P}\left[\sum_{k=1}^{n} X_k\leq a\right]\geq1-\mathbb{P}\left[\max\{X_1, X_2, \dotsc, X_n\}\leq a\right]$$
Which is the original inequality
About the commentary:
$$\sum_{k=1}^{n} X_k\leq a\Rightarrow\max\{X_1, X_2, \dotsc, X_n\}\leq a$$
Means:
$$\omega \in \Omega: \sum_{k=1}^{n} X_k(\omega)\leq a\Rightarrow\max\{X_1(\omega), X_2(\omega), \dotsc, X_n(\omega)\}\leq a$$
Which can be restated as:
$$\left\{\omega \in \Omega: \sum_{k=1}^{n} X_k(\omega)\leq a\right\}\subseteq \left\{\omega \in \Omega: \max\{X_1, X_2, \dotsc, X_n\}\leq a\right\} $$
Then the inequality comes up according to the probability measure property (monotonic):
$$A\subseteq B \Rightarrow \mathbb{P}(A) \leq \mathbb{P}(B)$$
A: No. Take i.i.d variables $X_1=\ldots=X_n=-1$ then $\max\{X_1,\ldots,X_n\}=-1$ and $\sum_{k=1}^nX_k=-k$. Then the inequality is false.
