Probabilistic Modus Tollens

Here's the rule that justifies the classical modus tollens inference:

[A => B] <=> [~B => ~A].

An example: if all kings have crowns, then not having a crown means one is not a king.

However, in real world, things aren't black-and-white. Therefore, a probabilistic equivalent of modus tollens would be nice. Here is one:

[P(A | B) >= P(A)] <=> [P(~B | ~A) >= P(~B)].

An example: if kings have crowns more often than other people, then not having a crown decreases (or doesn't increase) the probability of one being a king.

Proof is left as an exercise. (Hint: apply Bayes's rule and the rule of complementary probability.)