Probability and Given

Question

Question: in the following problem, can given be flipped?

$\Pr(A\cap F)=\Pr(A)\cdot \Pr(F\text{ given }A)=?$ Can I switch to get the same answer: $\Pr(A\cap F)=\Pr(A)\cdot P(A\text{ given }F)=?$

Answer 1

No, because it won't be true anymore. But think about the original formula as a template: $$\Pr(A\cap F)=\Pr(A)\cdot \Pr(F\mid A).$$ If you want $F$ to be "given", then you need to substitute $F$ for each $A$ of the formula and at the same time substitute $A$ for each $F$ of the formula. Note also that order doesn't matter for intersection. So you can get that $$\Pr(A\cap F)=\Pr(F\cap A)=\Pr(F)\cdot \Pr(A\mid F).$$