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I have trouble understanding one last step of my proof process. The problem is :

Let A,B,C be mutual indenpendent events, $P(C) > 0$ . Prove A, B are conditionally independent of C.

After translating to math language:

$$P(A \cap B | C) \Rightarrow P(A|C) \cdot P(B|C) $$

My prove:

$$P(A \cap B | C) \\ = \frac{P(A \cap B \cap C)}{P(C)} \\ = \frac{P(A) \cdot P(B) \cdot P(C))}{P(C)} \\ = P(A)\cdot P(B) $$

But I don't know how to interpret this as $P(A|C) \cdot P(B|C)$, Please help.

Yiffany
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  • Do you mean that $A,B,C$ are pairwise independent or mutually independent? – lulu Oct 25 '22 at 11:30
  • I think it's ,, are pairwise independent so $P(A \cap B \cap C)$ can be $P(A)P(B)P(C)$ – Yiffany Oct 25 '22 at 11:33
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    $P(A∩B∩C) =P(A)P(B)P(C)$ is not what pairwise independence means. Note your $\Rightarrow$s should be $=$. – Henry Oct 25 '22 at 11:34
  • Well, the claim is false if it is "pairwise" and true if it is "mutual". – lulu Oct 25 '22 at 11:34
  • I think you should review the definitions and basic properties. "mutual" implies "pairwise" and pairwise implies, among other things, that $P(A,|,C)=P(A)$. Should say: it's not unusual for people to write "independent" as shorthand for "mutually independent" but I think it's a bad idea and that it often leads to misunderstandings. – lulu Oct 25 '22 at 11:35
  • Hi @lulu, this is where I learn Conditional_independence events, and this I don't know what is pairwise or mutual. – Yiffany Oct 25 '22 at 11:40
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    here is a relevant question. You can find others. – lulu Oct 25 '22 at 11:42
  • @lulu, Thank you – Yiffany Oct 25 '22 at 11:44
  • Should say: mutual implies pairwise as a matter of definition. "mutual" means that any combination of events taken from the relevant collection is independent. – lulu Oct 25 '22 at 11:44
  • But still don't understand why ()⋅() = (|)⋅(|) – Yiffany Oct 25 '22 at 11:48
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    Try proving that if $A$ and $C$ are mutually independent then $P(A)=P(A\mid C).$ The proof is similar to what you've already done. – David K Oct 25 '22 at 11:52
  • If they are all pairwise independent you can say $P(A \cap B)=P(A)P(B)$ and $P(A \mid C)=P(A)$ and $P(B \mid C)=P(B)$. If they are mutually independent then you can say that and also say $P(A\cap B \mid C)=P(A \cap B)$ – Henry Oct 25 '22 at 12:34

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