Consider two independent random variables $(X,Z)$ where $Z$ is standard normal and $X$ has a pdf $f_X$. I want to see if we can somehow 'simplify' the following probability \begin{align} \mathbb{P}[ aX-Z \ge 0| X+Z] \end{align} for a given $a>0$.
For example, can this be written as an integral that only involves the joint of $f_{X,Z}$? Basically, how far can we simplify this expression or re-write it in a nice way.
I was thinking of writing this probability as follows \begin{align} \mathbb{P} \left[ \left[ \begin{array}{c} a \\ -1 \end{array} \right ] [X,Z] \ge 0 \mid \left[ \begin{array}{c} 1 \\ 1 \end{array} \right ] [X,Z]\right] \end{align} and then using some kind of mapping theorem, but I couldn't really do it.