I'm quite new to probability and I'm quite confused by a certain "operation" that is performed in my lecture notes. It's the operation of conditioning both sides of an equation on a certain event. For example, in my notes I have:
$$P(A\cap B)=P(A)P(B).$$
By conditioning on $C$ we obtain:
$$P(A\cap B\mid C)=P(A\mid C)P(B\mid C).$$
But how do we know that we can use this kind of conditioning on a particular equation? I can think of many examples when such conditioning would lead to errors. For example, rolling a die:
If we define:
$A$ as getting either $1$ or $2$,
$B$ as getting either $5$ or $6,$
and $C$ as getting $4$ of more, then it's true that:
$$P(A)=P(B).$$
But:
$$P(A\mid C)\neq P(B\mid C).$$
Could anyone help me understand this problem? Thank you in advance.