I can show that this identity is true using algebra.
We know if $$P(A \mid B) > P(A \mid B^c),$$ then $$ P(A\cap B) > P(A)P(B).$$
However, I am trying to understand my textbook's arugment using the law of total probabilty.
It states that since $$P(A) = P(A \mid B)P(B) + P(A \mid B^c)P(B^c),$$
then $P(A \mid B) > P(A \mid B^c)$ must imply that $P(A \mid B) > P(A)$.
I am lost.
Also this is my first question I am asking on here. Open to criticism on how I could ask better questions.