I am working to explore a property of conditional probability that I could use some assistance with. The general problem can be stated as
$$\Pr(a > b \mid z^*)$$
where $z = z^* + u$. So then
$$\Pr(a > b \mid [z-u])$$
In the second probability statement, is there a property that would allow for the term $u$ to distribute to the left of the $|$ or otherwise reduce.
In the current problem I am working with $u$ is unobserved. It has some expected values, such as $E(u)=0$ and $E(u'u) = \sigma^2$, but I do not know if that is helpful within this context.