A intersect B intersect C : Still confused I have the following question with A intersect B intersect C
I've broken down the formula below but I am stuck on $\mathbb{P}(B|\ A)$ below

$\begin{align}\mathbb{P}(A) ~=~& 0.4
\\[2ex]
\mathbb{P}(B) ~=~&0.2
\\[2ex]
\mathbb{P}(C) ~=~&0.05
\\[2ex]
\mathbb{P}(A \cap B \cap C)
~=~ & \mathbb{P}((A \cap B) \cap C) 
\\[1ex]
=~ & \mathbb{P}(C\mid A \cap B) \cdot \mathbb{P}(A \cap B) 
\\[1ex]
=~ & \mathbb{P}(C\mid A \cap B ) \cdot \mathbb{P}(B \mid A) \cdot \mathbb{P}(A)
\end{align}$

I'm confused by $\mathbb{P}(B \mid A) \cdot \mathbb{P}(A)$, because when I look for the formula to solve $\mathbb{P}(B\mid A)$ I'm brought back to the formula $\mathbb{P}(A \cap B)$. How do I solve for either $\mathbb{P}(B\mid A)$ or $\mathbb{P}(A \cap B)$ if they are same?
Any breakdown or help would be appreciated! 
 A: In spite of some gaps and confusion, maybe I can give you a start with some of this.
For simplicity of notation, I will adopt the frequently used convention of writing intersection as 'products'. That is, I'll write $A \cap B$ as $AB$, etc.


*

*As mentioned in a couple of comments, you need more information to give a specific 
numerical value for $P(ABC).$ It could be $0$ or $.05$ or anything between.
Do you understand these two possible boundary values?

*$P(ABC) = P((AB)C)$ because $ABC$ and $(AB)C$ mean the same thing. (Associative rule for set intersection.) 

*$P(C|AB)P(AB) = \frac{P(ABC)}{P(AB)}P(AB) = P(ABC),$
by the definition of conditional probability.

*Similarly the last product of three factors is also equal to $P(ABC).$
Can you write this out on your own?
Note: I have the feeling you are just beginning to study these ideas, and that you
have left something out, especially for the first item. Please fill in gaps or ask additional specific questions, if you need more explanation.
