I have two variables $X$ and $Y$. I want to find the following probability $$P(X>a(Y+c),X+Y>d)$$ where $a>0,c>0,d>0$. To find the solution I have done following steps $$E_Y[P(X>a(Y+c),X>d-Y)]$$ where expectation is with respect to $Y$. Now we can split the above expression in two parts depending on the event $a(Y+c)>d-Y$. Hence, we can write $$E_{Y|Y>\frac{d-ac}{a+1}}[P(X>a(Y+c))]P(Y>\frac{d-ac}{a+1})+E_{Y|Y<\frac{d-ac}{a+1}}[P(X>d-y)]P(Y<\frac{d-ac}{a+1})$$ I want to know if the above expression is right or the expression given below $$E_{Y}[P(X>a(Y+c))]P(Y>\frac{d-ac}{a+1})+E_{Y}[P(X>d-y)]P(Y<\frac{d-ac}{a+1})$$ The difference is the use of conditional expectation in the first expression while unconditional expectation in the second expression. In my own understanding I think the first expression is correct. However, I have seen in a paper that authors used second expression therefore I am quite confused. Please provide some insights over it. Thank you.
The image from the paper. 