Conditional probability

Events

Given two events $A$ and $B$ such that $P(B)>0$, the conditional probability of $A$, given $B$, is

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

Given the same constraints for $A$ and $B$, we can rearrange the above formula for another useful multiplication rule. Really, this is the same formula.

$$P(A\cap B)=P(B)\cdot P(A\mid B)$$

It is also equivalent to

$$P(A\cap B)=P(B\cap A)=P(A)\cdot P(B\mid A)$$