I'm reading Judea Pearl's "Book of Why" and although I find it really interesting (and potentially useful) I find the lack of explicit equations difficult to deal with. I want to know if I'm getting the back-door adjustment formula correct.

I understand that process for getting the list of confounders using the back-door criteria. So let's say I want to understand the effect of $X$ on $Y$ and I have a single confounder $Z$. As I understand it, the formulas are (assuming all the variables are binary):

$Pr[X|do(Y)] = Pr[X=1|Y=1,Z=0] \times Pr[Z=0] + Pr[X=1|Y=1,Z=1] \times Pr[Z=1]$

$Pr[X|do(!Y)] = Pr[X=1|Y=0,Z=0] \times Pr[Z=0] + Pr[X=1|Y=0,Z=1] \times Pr[Z=1]$

and the "effect" of $Y$ on $X$

$Pr[X|do(Y)] - Pr[X|do(!Y)]$

Are these correct? Also, which one is referred to as the "back-door adjustment formula"?