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I am asked to prove the following claim:

If $A$ and $B$ are separate events then $A$ and $B$ cannot be independent.

Intuitively, it is easy to see that the claim is true, since if $A$ occurs then $B$ must not have occurred (hence knowledge about $A$ gives complete knowledge about $B$, which is very strong form dependence).

Mathematically, $A$ and $B$ are independent if $P(A\cap B) = P(A)P(B)$. However, if $A$ and $B$ are separate, then there is nothing in the intersection of $A$ and $B$. Hence $P(A\cap B)=0$. However, $P(A)P(B)>0$ if $A$ and $B$ are possible events... So I have arrived in a contradiction? Is this sufficient to prove the claim?

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What is meant by separate? Disjoint? Do you mean $A \cap B = \emptyset$?

If so, then your claim is equivalent to:

If $A \cap B = \emptyset$, then $P(A \cap B) \ne P(A) P(B)$

To try to prove this, let us suppose $P(A \cap B) = P(A) P(B)$ but $A \cap B = \emptyset$.

As you pointed out, $P(A \cap B) = P(\emptyset) = 0$.

Then $P(A) P(B) = 0$ which means $P(A) = 0$ or $P(B) = 0$.

I don't really see any contradiction here unless $P(A) > 0$ and $P(B) > 0$.

You said $P(A)P(B) > 0$ if they are possible events. Are they? If not, I think your claim is false.

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  • $\begingroup$ It is said they are possible events in a probability space $(\Omega, \mathcal{F})$. With separate I meant disjoint, yes. $\endgroup$ – lisa Sep 18 '15 at 11:17
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    $\begingroup$ Is separate not a commonly used term? If one would draw a Venn diagram, I would find calling two non-overlapping blobs separate somehow intuitive. $\endgroup$ – lisa Sep 18 '15 at 11:20
  • $\begingroup$ Probability space is $(\Omega, \mathscr{F}, \mathbb{P})$ :P Anyway, where did you say that they are possible events? Okay $A, B \subset \Omega$, $A, B \in \mathscr{F}$, but does either mean $P(A), P(B) > 0$ ? Even if they are jointly exhaustive (perhaps that is an assumption you forgot to state?) ie$A \cup B = \Omega$, I don't see how that means $P(A), P(B) > 0$. I think separated means something different in math. I think the terminology is that the events are mutually exclusive (i.e. disjoint) $\endgroup$ – BCLC Sep 18 '15 at 11:22

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