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Roll $2$ dice.
Let $E$ be the event that the sum of the dice is $6$
Let $F$ be the event that the sum of the dice is $7$
Let $G$ be the event that the first die rolled is a $4$

$E$ and $G$ are dependent (since $P(E\cap G) \neq P(E)P(G)$ )
$F$ and $G$ are independent (since $P(F\cap G) = P(F)P(G)$ )

Intuitively, why is this true?

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    In neither cases are they causally independent, which I think relates to your intuition. But in the latter case they are uncorrelated.2012-12-10
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    @Lucas: You say that $F$ and $G$ are uncorrelated. What do you mean by that? I know that the correlation applies to random variables, but here it is about events. Also, $P(F\cap G) = P(F)P(G)$ means, by definition, that $F$ and $G$ are independent events. What do you mean that they are not "causally independent"?2016-09-17
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    @digital-Ink good question. In that comment I was language to highlight the distinction between causation and correlation - and avoid using the terms dependent independent. So I was using correlation loosely here (though you could assign numerical values to the events and speak of correlation properly) I understood it to be implicit in the question that the first intuition would be that neither are be dependent - because neither dice affects (causally) the other. The question of what causation is is complex. Here, by causally independent I mean something not physically affecting something else2016-09-17

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