TITLE: Conditional Probability P(A intersect B intersect C)
QUESTION [23 upvotes]: I'm curious what the breakdown of how the transition happens per the formula below.
I get how $P(A \cap B) = P(A\mid B)P(B)$ which is the famous conditional probability.
But am totally lost when there are three sets involved. Thanks!!
$$P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)$$

REPLY [20 votes]: It’s just a double application of the two-event formula, first thinking of $A\cap B$ as a single event:
$$\begin{align*}
P(A\cap B\cap C)&=P\big((A\cap B)\cap C\big)\\
&=P\big(C\mid(A\cap B)\big)P(A\cap B)\\
&=P\big(C\mid(A\cap B)\big)\Big(P(B\mid A)P(A)\Big)\\
&=P(A)P(B\mid A)P(C\mid A\cap B)\;.
\end{align*}$$

REPLY [3 votes]: It is the same thing but you use the same rule two times. 
$$P(A,B,C)=P(C\mid B,A)P(B,A)$$
Here assume $(A,B)=K$ then $P(A,B,C)=P(C\mid K)P(K)$ same with your rule. Then for the second case
$$P(A,B,C)=P(C\mid B,A)P(B,A)=P(C\mid B,A)P(B\mid A)P(A)$$
using again the same rule $P(B,A)=P(B\mid A)P(A)$.