Example of pairwise but not mutually independent

Question

Give an example of a probability space (Ω,Pr) and pairwise independent events A, B, and C which are not mutually independent.

This is my understanding of what pairwise independent events are: Events $A_1, A_2,..., A_k$ are pairwise independent if for all i,j, $A_i $ and $A_j$ are independent: $Pr(A_i\cap A_j) = Pr(A_i)Pr(A_j)$

Events $A_1, A_2,..., A_k$ are mutually independent if for all $I\subset 1,2,...,k, Pr(\bigcap ._{i\subset I} A_i) = \prod\limits_{i\subset I} Pr(A_i) $ so $ Pr(A_1)\cap Pr(A_2) ... Pr(A_k) = Pr(A_1)Pr(A_2)...Pr(A_k)$

Answer 1

I suggest @Ashley to refine your question, but meanwhile according to your title, I'll give an example of "pairwise but not mutually independent".

You toss a fair coin twice, let

$A_1$=Both tosses give the same outcome (HH or TT).

$A_2$=The first toss is a heads (HT HH).

$A_3$=The second toss is heads (TH HH).

We have:

$\mathbb P(A_1)=\mathbb P(A_2)=\mathbb P(A_3)=\frac{1}{2}$

$\mathbb P(A_1\cap A_2)=\mathbb P(A_1\cap A_3)=\mathbb P(A_2 \cap A_3)=\frac {1}{4}$

Thus, $\mathbb P(A_i \cap A_j)=\mathbb P(A_i)\mathbb P(A_j), \forall \,i,\,j \in \{1,2,3\},\, and \,i \ne j$

However, they are NOT mutually independent, by noticing that:

$\mathbb P(A_1\cap A_2\cap A_3)=\frac{1}{4}\ne \mathbb P(A_1)\mathbb P(A_2)\mathbb P(A_3)=\frac{1}{8}$