# 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

 $\displaystyle P(X_{r})=P(X_{g})$ $\displaystyle=P(X_{b})=\frac{1}{2},$ $\displaystyle P(X_{r}\cap X_{g})=P(X_{w})$ $\displaystyle=\frac{1}{4}=P(X_{r})P(X_{g}),$ but $\displaystyle P(X_{r}\cap X_{g}\cap X_{b})=\frac{1}{4}$ $\displaystyle\neq\frac{1}{8}=P(X_{r})P(X_{g})P(X_{b}).$
Title example of pairwise independent events that are not totally independent ExampleOfPairwiseIndependentEventsThatAreNotTotallyIndependent 2013-03-22 13:38:51 2013-03-22 13:38:51 bbukh (348) bbukh (348) 6 bbukh (348) Example msc 60A05