Pairwise independence does not imply independence Find an ex

Pairwise independence does not imply independence Find an example of three events that are pairwise independent but not independent.

Solution

Proof

Consider throwing a fair four-sided die.

This gives us an event space {1,2,3,4}, with each points are equally likely to occur:

With probability = 1/4

Consider the set of events:

{A,B,C}

where:

A={1,2}, B={1,3}, C={1,4}

We have that:

Pr(A)=Pr(B)=Pr(C)=1/2

We also have that:

Pr(AB)=Pr(AC)=Pr(BC) = Pr({1}) = 1/4

Thus:

Thus the events A,B,C are pairwise independent.

Now, consider:

Pr(ABC)=Pr({1})=1/4

But:

Pr(A)Pr(B)Pr(C)=1/8 Pr(ABC)=1/4

So, although A,B,C are pairwise independent, it is not independent.