# why conditional probability has a given formula

I'm taking an intro to statistics class and currently we are covering conditional probability. The chapter explained what it is quite well, but it didn't explain how the formula for it was derived, i.e it didn't explain explain how

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

was derived.

I would greatly appreciate if someone could show the reasoning as well as derivation of this formula.

When we reduce the sample space to $B$ then the only way for $A$ to occur is if $B$ does as well, hence the $P(A \cap B)$. And the size of this event with respect to the new sample space is $P(A \cap B) / P(B)$, or the fraction of $B$ taken up by $A \cap B$.