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

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?

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