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

Question

1768 views around the world

You can reuse this answer
Creative Commons License

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.