2.1.14 problem 16

$P\left(A\right)=P\left(B\right)P\left(A\right|B)+P(B^c\left)P\right(A\left|B^c\right)$.

Given $P\left(A\right|B)\le P(A)$, if $P\left(A\right|B^c)<P(A)$, then the right hand side of the equation above is strictly less than the left hand side, and we have a contradiction.

We can intuitively think of this problem as asking "How likely is X to be elected as president?" and hearing "It depends" in response. The implication is that there exists some latent event (major states vote against X) that reduces the chances of X getting elected, and if we know that the former does not occure, the chances of X getting elected improve.