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

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

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  • $\begingroup$ So the ordering is unimportant, right? I just want to confirm this! $\endgroup$
    – Martin08
    Jan 26 '11 at 23:30
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    $\begingroup$ @Martino8: Yes it is unimportant. $\endgroup$ Jan 26 '11 at 23:40

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