# Conditioning on more than one event

#### Buckeye

##### Member
Lately, I've been studying conditional probability. I'm working with the following theorems:

1. P(A|B)=P(A,B)/P(B) (the comma represents intersection)
2. P(A1,A2,...,An)=P(A1)P(A2|A1)...P(An|A1,A2,...An-1)
3. P(A|B)=[P(B|A)P(A)]/P(B) (Bayes' Rule)

I'm curious about conditioning with more than one event:

P(B|A1,A2,...,An) for n events.

In particular, I don't know the stipulations for the A1,A2,...,An
As I understand conditional probability, I am updating prior probability with the evidence I collect. Can someone give me a formal definition for conditioning with more than one event and some working theorems?

#### Dason

I think you need to be a bit more specific with your question. I'm not quite clear on what you're asking. There really are no stipulations on the events that you condition on.

#### Dason

Does it help if I tell you that typically [math]A_1, A_2, \ldots, A_n[/math] is shorthand for [math]A_1 \cap A_2 \cap \ldots \cap A_n[/math]? Just think of it like any other set. Or do something like define [math]A = A_1 \cap \ldots \cap A_n[/math] and then apply what you know.