Conditional Probability

The Probability of $A$, $P(A)$, given $B$ has occurred ($P(B)>0$)

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

Simple Sample Space

If the Sample Space is a Simple Sample Space, i.e. where $\forall \omega \in S,P(\{\omega \})=\frac{1}{|S|}$, and all of the above is already true, then the following is valid.

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

Disjoint Events

$$\text{djt(A,B)}⟹P(A|B)=\frac{0}{P(B)}=0$$

Independent Events

$$\text{djt(A,B)}⟹P(A|B)=\frac{P(A)P(B)}{P(B)}=P(A)$$