Is it true that non mutually exclusive events can only be independent events?

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Answer 1 · 2017-08-09T20:13:25

No. It is not always true. See explanation.

First we have to look at the definitions.

Two events $A$ and $B$ are independent if and only if $P(AnnB)=P(A)xxP(B)$

Two events are mutualy exclusive if their product is empty set
(i.e. $P(AnnB)=0$ )

Let's consider 2 events with throwing a 6 sided die.

$A$ - a result is not more than $5$ and

$B$ - a result is more than $4$

These 2 events are not mutualy exclusive because their product is:

$AnnB$ - the result is more than $4$ and not more than $5$ .

so: $AnnB={5}$ , $P(AnnB)=1/6$

Now let's check if the events are independent:

$A={1,2,3,4,5}$ , $B={5,6}$ , so:

$P(A)=5/6$ , $P(B)=2/6=1/3$ , so:

$P(A)xxP(B)=5/6xx1/3=5/18$

$P(A)xxP(B) != P(AnnB)$ , so the events are not independent.

Conclusion:

Example events are not mutualy exclusive and not independent, so it is a counter-example against hypothesis stated in the question.