Conditioning on Multiple Events

Definition

For events, $A,B,C$ the probability of $A$ given $B$ and $C$ is denoted as: $P(A|B,C)=P(A|B\cap C)=\frac{P(A\cap B\cap C)}{P(B\cap C)},P(B\cap C)>0$

Disjoint Generalization

If $A,B,C$ are mutually independent, then:

• $P(A|B,C)=P(A)$ i.e, conditioning on any collection of events does not affect the probability of intersection on any other events.