Two questions about inequality

Question

Problem

1. Suppose $\alpha \in [0,1], A\geq a$, then we have the following argument $$ P[X\geq a] \leq \alpha \Rightarrow P[X\leq a] \geq 1-\alpha \Rightarrow P[X\leq A] \geq 1-\alpha\Rightarrow P[X\geq A]\leq \alpha $$ So is this argument correct?

2. By union bound, we have $$ P[\vert X\vert \geq t] \leq P[X\geq t] + P[X\leq -t] $$ Moreover $$ P[X-\mathbb{E}[X]\geq t] \leq \frac{\delta}{2}\\ P[\mathbb{E}[X]-X\geq t] \leq \frac{\delta}{2} $$ Is there any way we could bound $P[\vert X\vert \geq t]$ using $\delta$?

Answer 1

1. You can skip the middle two steps by noting that $\{X \ge A\} \subseteq \{X \ge a\}$ so $P(X \ge A) \le P(X \ge a) \le \alpha$.

2. When $E[X] = 0$, the last two inequalities imply $P(|X| \ge t) \le \delta$ by your union bound. More generally, the last two inequalities imply $P(|X - E[X]| \ge t) \le \delta$.