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[parent] example of pairwise independent events that are not totally independent (Example)

Consider a fair tetrahedral die whose sides are painted red, green, blue, and white. Roll the die. Let $ X_r, X_g, X_b$ be the events that die falls on a side that have red, green, and blue color components, respectively. Then

$\displaystyle P(X_r)=P(X_g)$ $\displaystyle =P(X_b)=\frac{1}{2},$    
$\displaystyle P(X_r \cap X_g)=P(X_w)$ $\displaystyle =\frac{1}{4}=P(X_r)P(X_g),$    

but


$\displaystyle P(X_r \cap X_g \cap X_b)=\frac{1}{4}$ $\displaystyle \neq \frac{1}{8}=P(X_r)P(X_g)P(X_b).$    



"example of pairwise independent events that are not totally independent" is owned by bbukh.
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Cross-references: color, events, sides

This is version 3 of example of pairwise independent events that are not totally independent, born on 2003-05-26, modified 2005-12-30.
Object id is 4299, canonical name is ExampleOfPairwiseIndependentEventsThatAreNotTotallyIndependent.
Accessed 3539 times total.

Classification:
AMS MSC: 60A05 (Probability theory and stochastic processes :: Foundations of probability theory :: Axioms; other general questions)
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