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Suppose $A $ and $B$ are independent events. For an event $C $ such that $P(C) > 0$ , prove that the event of $A$ given $C $ is independent of the event of $B$ given $C $

We have A and B are independent so $P (AB) = P (A) \cdot P (B) $

We need to show that $P ((A\mid C)\cap (B\mid C)) = P (A\mid C)\cdot P (B\mid C)$

My procedure was like this $$P ((A\mid C)\cap (B\mid C)) = P (AB \mid C) $$

I played arount to get this $$ \frac {P (AC)}{P (C)} \cdot \frac {P (B \mid AC)}{P (B \mid C)} $$ Now the first part gives us $P (A \mid C) $ . I couldn't get from the second part the missing part which is $P (B \mid C) $.

Is my procedure correct? If so, how can I find the second part?

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  • $\begingroup$ $A|C$ is not an event and $(A|C)\cap (B|C)$ has no meaning. $\endgroup$ – Kavi Rama Murthy Mar 15 at 8:16
  • $\begingroup$ Why it's not an event? Doesn't it have a probability? $\endgroup$ – Noussa Mar 15 at 8:17
  • $\begingroup$ Did you just make up the question, or it's from somewhere? I don't think it's correct. $\endgroup$ – Minus One-Twelfth Mar 15 at 8:28
  • $\begingroup$ It's from my textboot. Introduction to probability Scheaffer $\endgroup$ – Noussa Mar 15 at 8:29
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    $\begingroup$ I think you should edit the question using exactly the words and symbols used by the author. $\endgroup$ – Kavi Rama Murthy Mar 15 at 8:48

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