Extending conditional probability to more than 2 events

Question

Quick question, In the context of conditional probability if we can define $$P(A\cap B) = P(A|B) P(B) $$ Then what is a similar way to write $$P(A\cap B \cap C)?$$

Answer 1

Well, since

$$P(A|B) = \frac{P(A\cap B)}{P(B)}$$ by definition, you can show that $$ P(A\cap B) = P(A|B)P(B).$$ In a similar fashion you can show that

$$ P(A\cap B \cap C) = P(A|B\cap C)P(B|C)P(C).$$ I hope this helps.