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On Wikipedia on page relative to Conditional probability in section Statistical independence it is written following

It should also be noted that given the independent event pair [A B] and an event C, the pair is defined to be conditionally independent if the product holds true:[17]

$$P(AB\mid C)=P(A{\mid}C).P(B{\mid}C)$$

Is that equivalent to say that

If A and B are two mutually independent events, than

$$P(A\cap B\mid C)=P(A{\mid}C).P(B{\mid}C)$$

Is that true ?

How can I prove the last formula ? Wikipedia is citing her source, but cited book are not readable online !

PS: I have "corrected" Wikipedia sentence today. Perhaps that my correction is incorrect !

schlebe
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1 Answers1

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Last formula is not true in general. For example taking $$\Omega =\{ 1,2,3,4 \}, A = \{1, 2\}, B=\{ 2,4 \}, C=\{1,2,4\},$$ with classical probability give you pair of independent events $A, B$. But $$P(A\cap B| C) = \frac 1 3$$ and $$ P(A| C)P(B| C) = \frac 2 3 \cdot \frac 2 3 = \frac 4 9.$$

This shows that your definition of conditionally independent events actually introduces something different to regular independence. It could be thought of as independence in restricted space $C$.

Esgeriath
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  • I'm not sure that your answer is correct because in your example A and B are NOT independant. You (or me) confuse event intersection with set intersection ! For me A∩B is the probability that A occurs in omega set and than B occurs in omega set. For you, A∩B is the probability that intersection of A and B sets occurs in omega set. This is not the same thing. I'm not sure that my syntax (A∩B) is correct for event ! – schlebe Dec 09 '22 at 12:45
  • Intersection of sets (events) is exactly the event of "both events happening". I think your confusion might come from not understanding definition of probability space properly. Also, as I read your question again, my answer could have been off-topic. But comment of John Douma wasn't. – Esgeriath Dec 09 '22 at 16:08