Let $E, F$, and $G$ be three events. If $P[E|F] ≥ P[G|F]$, is it sufficient to conclude that $P[E] ≥ P[G]$? What additional relationship needs to be known between $P[E|F_c]$ and $P[G|F_c]$?
For the first question, I have tried to expand the equation using conditional probability formula but yields no result. Then i add a condition that they are mutually independent such that their intersection is equal to their product and proven the inequality
My question is that can it be proven without the mutually independent condition and can i use the same condition and method to prove the second question that is: probability of $E$ given $F^\complement$ is greater than equal to probability of $G$ given $F^\complement$?
