# Example of two non-mutually exclusive events that are dependent?

(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?

• Roll a die. Let $E$ be the event the roll is $1$, and let $F$ be the event the rolls is either $1$ or $2$. Apr 19, 2019 at 22:14
• 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 $P(F|E)=1\neq P(F)$. Apr 19, 2019 at 22:46
• Your statements are not precise. The events $\emptyset$ and the entire sample space $\Omega$ are independent and mutually exclusive. Apr 19, 2019 at 23:13