Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence

Question

Would someone please advance/discuss some real-life situations falsities $1, 2$?
I'd like to intuit why these are false. As a neophyte, since I still need to compute the probabilities for the examples in the two answers to apprehend them, I haven't naturalised these concepts yet.
Thus, I beg leave to ask about other real-life examples which require less or no computations.

I tried and would appreciate less numerical examples than http://notesofastatisticswatcher.wordpress.com/2012/01/02/pairwise-independence-does-not-imply-mutual-independence/ and http://econ.la.psu.edu/~hbierens/INDEPENDENCE.PDF, and Examples 1.22 and 1.23 on P39 of Introduction to Pr by Bertsekas.

$1.$ Pairwise Independence $\require{cancel} \cancel{\implies}$ (Mutual) Independence.

$2.$ Pairwise Independence $\require{cancel} \cancel{\Longleftarrow}$ (Mutual) Independence.

P38 defines (Mutual) Independence : For all $S \subseteq \{1, ..., n - 1, n\} $ and events $A_i$, $Pr(\cap_{i \in S} A_i) = \Pi_{i \in S} A_i.$

Answer 1

The usual (and perhaps the most basic) example is to throw two fair coins and to consider the three following events:

• "The first coin shows heads"
• "The second coin shows heads"
• "The two coins agree"

Then, the probability of each of these events is $.5$, the probability of their intersection is $.25$ and the probability of each intersection of two of them is also $.25$.

Thus, they are not independent and they are pairwise independent.

Answer 2

Suppose three guys each toss a (fair) coin. The events "A and B match", "A and C match", "B and C match" are pairwise independent; the three events are not mutually independent.