How to prove P(A ∩ B ∩ C) = P(A|B ∩ C)P(B|C)P(C)?

Question

How to prove P(A∩B∩C) = P(A|B∩C)P(B|C)P(C)?

I've tried to approach this a couple times, but always end up getting stuck :(

Any help is appreciated! :)

Answer 1

Since $P(A|B\cap C) = P(A\cap B\cap C)/P(B\cap C)$, we have

$$P(A|B\cap C)P(B\cap C) = P(A\cap B\cap C).$$

Since $P(B|C) = P(B\cap C)/P(C)$,

$$P(B|C)P(C) = P(B\cap C).$$

Therefore

$$P(A|B\cap C)P(B|C)P(C) = P(A\cap B\cap C).$$