Let A and B be two events of an experiment.

Given a statement:
A implies B but B does not imply A (or simply A implies B).

A: Otaku.
B: Watch Anime.
That is: Otaku watches Animes. But those who watch Anime may not be an Otaku.

In Set Theory, we may say A is inside B but B is not inside A. Or A \subset B.

To put them into conditional probability:

\Pr(B|A)=\frac{\Pr(B \cap A)}{\Pr(A)}=\frac{\Pr(A)}{\Pr(A)}=1

\Pr(A|B)=\frac{\Pr(A \cap B)}{\Pr(B)}=\frac{\Pr(A)}{\Pr(B)}\leq 1


Thus, the statement: “A implies B but B does not imply A” may be written as:

\Pr(B|A)=1 and \Pr(A|B) \leq 1.




Gaia disagree with my misuse of Mathematics notation.