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# 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 | ||||

$\displaystyle P(X_{r}\cap X_{g}\cap X_{b})=\frac{1}{4}$ | $\displaystyle\neq\frac{1}{8}=P(X_{r})P(X_{g})P(X_{b}).$ |

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