TITLE: Why use A|BC and not ABC?
QUESTION [6 upvotes]: Currently I'm reading Probability Theory The Logic of Science and there it's not really clear for me what exactly $A|BC$ means.
So I have two questions:
Is this a general notation?
Is $A|BC$ equal to $ABC$? If it is — that means it's just for the sake of order/consecution, like $A(BC)$ (but according to the book it's not so)?

REPLY [5 votes]: $A|BC$ does not mean much 
but the conditional probability $P(A \mid B,C)$ does, perhaps being read as the probability that event $A$ occurs given that the events $B$ and $C$ both occur; 
This is not the same as the joint probability $P(A , B,C)$ which is the probability that all three events occur
As an example, suppose you throw three standard dice, and the events are: 

$A$ all three dice show the same value
$B$ the sum of the three values is greater than or equal to $16$ 
$C$ the sum of the three values is even 

There are $6^3=216$ equally probable ways of throwing the dice, of which $6$ satisfy $A$, $10$ satisfy $B$, $108$ satisfy $C$, $7$ satisfy $B$ and $C$, and $1$ satisfies all three of  $A$ $B$ and $C$.
So $P(A, B,C) = \frac{1}{216}$ but $P(A \mid B,C) = \frac{1}{7}$, rather different
You can show the relationship between the conditional probability and the joint probability with $$P(A \mid B,C)\, P(B,C) = P(A, B, C)$$