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(1)If two events are independent that implies that they are "non-mutually exclusive".

Then by using logic transposition,

Non "non-mutually exclusive" events implies they are "not independent"

That's to say that if two events are mutually exclusive they are dependent.

But by logic, from (1), if two events are "non-mutually exclusive" then they don't have to be necessarily independent. So there are non-mutually exclusive events that are dependent.

Can you give an example of it?

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    $\begingroup$ Roll a die. Let $E$ be the event the roll is $1$, and let $F$ be the event the rolls is either $1$ or $2$. $\endgroup$ – Mike Earnest Apr 19 at 22:14
  • $\begingroup$ Just compute $P(E\cap F)$, and see it is not equal to $P(E)\times P(F)$. Alternatively, knowing that $E$ occurred tells you that $F$ occurred, so $P(F|E)=1\neq P(F)$. $\endgroup$ – Mike Earnest Apr 19 at 22:46
  • $\begingroup$ Your statements are not precise. The events $\emptyset$ and the entire sample space $\Omega$ are independent and mutually exclusive. $\endgroup$ – Kabo Murphy Apr 19 at 23:13
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The extreme example of a "non-mutually exclusive" set of events would be two mutually exclusive events. Thus you can just take the event of flipping a head on the first flip of a freshly minted coin and flipping a tail on the first flip of a freshly minted coin; even though these have the same probability (assuming the coin is fair), the events are mutually exclusive, and thus dependent on each other.

Did I read that question correctly and get you your answer?

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