Let $(X_1, X_2)$ be a randomly chosen pair out of $\{1,2, \ldots, 20\}$ (draw without repetition). Are both events $E_1:=\{X_1 \geq 8\}$ and $E_2:=\{X_2 \geq 12\}$ positive or negative correlated. Are they independent?
$P(E_1\cap E_2) = \frac{9}{20}$
and
$P(E_1) \cdot P(E_2) = \frac{13}{20} \cdot \frac{9}{20} = \frac{117}{400}$
on the basis of
$P(E_1 \cap E_2) > P(E_1)\cdot P(E_2)$
-> positiv correlated?
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We expect that both events $(E_1\text{ and }E_2)$ happen at the same time (the pair is chosen in one draw).