Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is the order of random variables important in the chain rule? I mean, is this true: P(A,B,C) = P(A)P(B|A)P(C|A,B) = P(C)P(B|C)P(A|B,C) = P(C,B,A)? If it is, what is the meaning of such order? Thank you.

share|cite|improve this question
up vote 1 down vote accepted

$P[A \cap B \cap C] = P[(A \cap B) \cap C] = P[(A \cap B)|C]P(C) = P[C|A \cap B]P[A \cap B]$. Then you can rewrite $P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$.

These are all useful. Suppose you want to find $P(A \cap B)$. Well $P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$. But suppose you only know $P(B|A)$. Then $P(B|A)P(A)$ is more useful.

share|cite|improve this answer
So the ordering is unimportant, right? I just want to confirm this! – Martin08 Jan 26 '11 at 23:30
@Martino8: Yes it is unimportant. – PEV Jan 26 '11 at 23:40

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.