Expressing one conditional probability in terms of another conditional probability with one more term in the conditional

Question

With all the variables $F', F, A$, and $Y$ being binary $\in\{0,1\}$, how do I express the probability expression below $$ Pr[F'=1|A=0,Y=0] $$ in terms of the following expressions: $$ Pr_{1,0} = Pr[F'=1|F=1,A=0] $$ $$ Pr_{1,0} = Pr[F'=1|F=0,A=0] $$ and $Pr[F=f|A=a,Y=y]$, where $f, a, y$ can be any combinations of $\{0,1\}$

Answer 1

I doubt that you can without more information. You can say

$$\Pr[F'=1\mid A=0,Y=0] $$ $$= \Pr[F'=1, F=0\mid A=0,Y=0]+\Pr[F'=1,F=1\mid A=0,Y=0] $$ $$=\Pr[F'=1\mid F=0,A=0,Y=0]\Pr[F=0\mid A=0,Y=0]$$ $$+\Pr[F'=1\mid F=1,A=0,Y=0]\Pr[F=1\mid A=0,Y=0] $$

but you do not know what $\Pr[F'=1\mid F=0,A=0,Y=0]$ or $\Pr[F'=1\mid F=1,A=0,Y=0]$ are.

If they were equal to $\Pr[F'=1\mid F=0,A=0]$ and $\Pr[F'=1\mid F=1,A=0]$ respectively then you could proceed, but there seems no basis to believe that