Definition of Conditional Probability

Definition

Let

• $B,A$ : Events
• $P(A)>0$

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

Then we say $P(B|A)$ is the conditional probability of $B$ given $A$

Moreover,

1. $P(B|A)\ge 0$
2. $P(A|A)=1$
3. If $B_1,\dots ,B_n$ are mutually exclusive events, then $P({\cup }_{n=1}^{\infty }{B}_n|A)={\sum }_{n=1}^{\infty }P(B_n|A)$