Order of operations in statistics How should I understand P(a|b,c)?  Is it P() of a given b and c, or P() of a given b, and then c?  How can I flip the order inside the brackets?  Thanks.
 A: Everything to the left of the $\mid$ is the event whose conditional probability is being talked about; everything to the right of the $\mid$ 
is the conditioning event, the one that we assume has occurred. Commas
generally mean intersection. Thus, $P(A\mid B, C)$  is the same
as $P(A\mid B\cap C)$ or $P(A\mid (B\cap C))$, the conditional probability of the event $A$
conditioned on the occurrence of the event $B\cap C$ or
$(B\cap C)$ depending on how nitpicky you are about parentheses.
In fact, it is best to be nitpicky about parentheses when one is
starting to learn the subject (or when one is teaching the subject
to beginners) because many beginners think of 
$P(A \mid B \cap C)$ as meaning "the probability that
both the events (i) $A$ given $B$, and (ii) $C$, have occurred."
There is no event called "$A$ given $B$" that we can intersect
with $C$; there is only the event $A$,
and we can talk of the unconditional probability that $A$ occurred,
or of the conditional probability that $A$ occurred. Here, we
are considering the conditional probability of $A$, and the
conditioning  event in this instance is $B\cap C$. 
That vertical bar
is a bright shining line that separates the conditioned event from
the conditioning event.
A: Order doesn't matter
Order doesn't matter in this setting, so there isn't any order of operations to worry about.  Explicitly:
$$P(a \mid b, c) = P(a \mid c, b)$$
This is because the AND logical concept doesn't depend on order.  Consider the statement "It is Wednesday AND I am a student". This is an equivalent logical statement to "I am a student AND it is Wednesday"
Let's say $A$ represents whether or not I am currently in the library.  Then $P(A \mid \text{student AND Wednesday})$ is intuitive no different than $P(A \mid \text{wednesday AND student})$
