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