example of pairwise independent events that are not totally independent

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

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

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

but

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

Title	example of pairwise independent events that are not totally independent
Canonical name	ExampleOfPairwiseIndependentEventsThatAreNotTotallyIndependent
Date of creation	2013-03-22 13:38:51
Last modified on	2013-03-22 13:38:51
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	6
Author	bbukh (348)
Entry type	Example
Classification	msc 60A05