Suppose I have a random variable $W$ that can take values $w_1,w_2,w_3$ with probability $p_1,p_2,p_3$ with $p_1+p_2+p_3=1$ (exhaustive) mutually exclusive.
What is the difference between the following statements,
$$\mathbb P_W[\exists z \; s.t. \;Wz \geq c, |z| \leq 1] \tag{1}$$
and
$$\exists |z| \leq 1 \; \mathbb P_W[Wz \geq c] \tag{2}$$
Clearly there is a change in the order of quantifiers. But it is not clear what these events signify and how we can use the union bound to upper bound the probability of first expression using the second expression.
Here , (1) is meaningful , though a little round-about.
While (2) is not meaningful , even if we make minor improvements.
We have to avoid $\exists |z|$ & instead use $\exists z$ to make it Correct :
$$\exists z \; s.t. |z| \leq 1 \land \mathbb P_W[Wz \geq c] \tag{2A}$$
Even then , it says that there is some $z$ such that "Some Probability or Some Number" ????
Not meaningful !
We have to make it Complete.
$$\exists z \; s.t. |z| \leq 1 \land \mathbb P_W[Wz \geq c] = 0.5 \tag{2B}$$ $$\exists z \; s.t. |z| \leq 1 \land \mathbb P_W[Wz] \geq c \tag{2C}$$ $$\exists z \; s.t. |z| \leq 1 \land \mathbb P_W[Wz \geq c] \geq 0.5 \tag{2D}$$
[[ Whether $ s.t. $ is Implicit or Explicit , it is still there in the Statement ]]
Even with that Correction+Completion , (2) is only a Predicate , which can be either true or not true ( Depending on $W$ & $c$ & ETC & whether $z$ Exists or not ) : It is not a Probability value which we can use to calculate (1).