Why use A|BC and not ABC? 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)?
 A: $A|BC$ 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 $$P(A \mid B,C)\, P(B,C) = P(A, B, C)$$ 
A: In a probability theory course $x|y$ usually means "x given y". Elsewhere  $x|y$ might mean "x divides y" or "the set of x such that y".
$x|y$, or "x given y" might come up in context as follows: Let
$x = $ Your car runs well
$y = $ Your car is on fire
Then $P(x|y) \leftrightarrow$ the probability that your car runs well given that your car is on fire.
$\big(P(x|y)$ is at or trending towards zero$\big)$
This notation will be important when studying conditional probability (for example, Bayes' Theorem or the Law of Total Probability)
