I need to reason about the probability that a conditional probability is large, and I'm not sure where to start. Specifically, let $E$ be an event, and $Z$ be a random variable. Let $$ \mathcal A=\{z \in \operatorname{support}(Z) : \Pr(E|Z=z)\geq \alpha\}. $$ I want to reason about $\Pr(Z \in \mathcal A)$.
In sloppier notation: $$ \Pr(Z:\Pr(E|Z) \geq \alpha). $$
Is there a name for this kind of expression, any rules for rewriting it in a different form, or general tricks for dealing with an expression of this form?
To enlighten the meaning of $$Pr(Pr(E|Z) \geq \alpha)$$ let' see an example.
Let $$\Omega=\{1,2,3,4\}$$
with $P(\{1\})=\frac12$ and $P(\{i\})=\frac16$ for $i=2,3,4$ and $E=\{2,3\}$, and let $Z(\omega)=u$ for $\omega\in \{1,2\}$ and $v$ for the rest, finally let $\alpha=\frac13$.
Now $Pr(E\mid Z)$ is a random variable:
$$Pr(E \mid Z)(\omega)=$$ $$=\frac{Pr(E\cap \{\omega\})}{Pr(\{\omega\})}=$$ $$= \begin{cases} 0,&\text{ if }& \omega=1,4\\ 1,&\text{ if }& \omega=2,3 \end{cases}.$$
So $$Pr(Pr(E\mid Z)>\alpha)=\frac13.$$