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).