Example of an experiment in which A, B, C are independent, but not pairwise independent

Question

Can somebody give an example of process in which we have at least three events A, B, C and:

P(A ∩ B ∩ C) = P(A) * P(B) * P(C)

But A, B, C are not pairwise independent

Answer 1

Consider two independent rolls of a fair six-sided die, and the following events:

$A$ = {First roll is 1, 2 or 3},

$B$ = {First roll is 3, 4, or 5} ,

$C$ = {The sum of the two rolls is 9}.

Then, $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{2}$ and $P(C)=\frac{4}{36}$.

It can easily be seen that $P(A \cap B \cap C)=\frac{1}{36} = P(A)P(B)P(C)$

However, $P(A \cap B)=\frac{1}{6} \neq \frac{1}{2} \cdot \frac{1}{2} = P(A)P(B)$,

$P(B \cap C)=\frac{1}{12} \neq \frac{1}{2} \cdot \frac{4}{36} = P(B)P(C)$ and

$P(C \cap A)=\frac{1}{36} \neq \frac{4}{36} \cdot \frac{1}{2} = P(C)P(A)$.

Answer 2

If you are interested in a counterexample using events whose probabilities are nonzero, consider the following:

Roll a fair 8-sided die with sides labeled $1,2,\dots,8$.

Let $A$ be the event the die shows one of the numbers $1,2,3,4$. Let $B$ be the event the die shows one of the numbers $1,2,3,5$ and let $C$ be the event the die shows one of the numbers $1,6,7,8$.

It should be clear that $Pr(A)=Pr(B)=Pr(C)=\frac{1}{2}$, it should be clear that $Pr(A\mid B)=\frac{3}{4}\neq Pr(A)$ and also it should be clear that $Pr(A\cap B\cap C)=\frac{1}{8}$

Thus, we have here an example of three events where the intersection of all three is equal to the product of the respective probabilities of the three despite them not being pairwise independent.

As pointed out in the comments above, for a collection of events to be mutually independent, it requires that the set of them and any subset of them has the property that the probability of their intersection is the product of their probabilities. Despite in the above example us having $Pr(A\cap B\cap C)=Pr(A)\times Pr(B)\times Pr(C)$, we do not call these events independent since they fail to be pairwise independent.