Can someone please give me a reference to an (simple, realworld, i.e. not constructed) example of a discrete probability space such that there are three events in it that are pairwise independent but all three together are not independent (although I wouldn't mind, if someone would give me the example as an answer).
"Pairwise independent" is weaker that "independent"
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0@user26698: I've voted up your comment, but in general you'll get better answers if you mention what you've tried (e.g. "I looked at wikipedia and found the example artificial".) – 2014-02-11
2 Answers
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The standard example involves tossing $2$ fair coins. For a more symmetrical example, toss $3$ fair coins. Let $A$ be the event Toss $1$ and Toss $2$ give the same result, $B$ be the event Toss $2$ and Toss $3$ give the same result, and $C$ the event Toss $3$ and Toss $1$ give the same result.
We have $\Pr(A)=\Pr(B)=\Pr(C)=\frac{1}{2}$ and $\Pr(A\cap B)=\Pr(B\cap C)=\Pr(C\cap A)=\frac{1}{4}$.
However, it is clear that $A$, $B$, and $C$ are not mutually independent.
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http://en.wikipedia.org/wiki/Independence_%28probability%29#Pairwise_and_mutual_independence ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$