Expanding expression using Conditional Probability

Question

I have come across the following equivalency:

p(A, B|C) = p(A, B, C) / P(C)

I am trying to understand why/how the left side expands to the right side.

My attempts:

p(A, Z) = p(A|Z)/P(Z) = P( A | (B|C) ) / P( (B|C) )

But I have no idea what to make of that or if its even remotely on the right track...

Answer 1

Instead of making a variable to represent $B \mid C$, try to make a variable represent the event $A \wedge B$ (i.e., "$A$ and $B$")

Let $Z := A \wedge B$.

$$p(A,B \mid C) = p(Z \mid C) = \frac{p(Z,C)}{p(C)} = \frac{p(A,B,C)}{p(C)}$$

Answer 2

Exactly. It's just a matter of correctly placing parentheses.

Consider the question as:

$$ P((A\cap B) | C) $$ and then the usual law of conditional probability applies where

$$ P((A\cap B) | C) = \frac{P(A \cap B \cap C )}{ P(C)}$$