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Sorry for this simple question, I'm a first year student, it's really basic but I don't know how to answer The question is: A random experiment is conducted which has sample space $\Omega = \{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$. Assume that all elementary events are equally likely to occur. Let the event $A = \{14,15,16,17,18,19,20 \}$. Let $F$ be an event such $P(F)= 0.25$. Explain why, for this experiment, $A$ and $F$ cannot be independent.

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  • $\begingroup$ May be it should be $\mathbb{P}(F) = 0.75$? $\endgroup$ – Andrei Kulunchakov Dec 2 '15 at 7:24
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Note that

$$ \frac{|A \cap F|}{20} = P(A \cap F)$$

If $A$ and $F$ were independent, then $$P(A \cap F) = P(A) P(F) = \frac{7}{80}$$ which implies $|A \cap F| = \frac{7}{4}$ which is absurd.

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