# Something in My Mind

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 $B$).

e.g.
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.