Out of the students in a class, $60 \%$ are geniuses, $70 \%$ love chocolate, and $40 \%$ fall into both categories. Determine the probability that a randomly selected student is neither a genius nor a chocolate lover.
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Let $G$ and $C$ be the events that the chosen student is a genius and a chocolate lover, respectively. We have $\mathbf{P}(G)=0.6, \mathbf{P}(C)=0.7$, and $\mathbf{P}(G \cap C)=0.4$. We are interested in $\mathbf{P}\left(G^{c} \cap C^{c}\right)$, which is obtained with the following calculation:
$\mathbf{P}\left(G^{c} \cap C^{c}\right)=1-\mathbf{P}(G \cup C)=1-(\mathbf{P}(G)+\mathbf{P}(C)-\mathbf{P}(G \cap C))=1-(0.6+0.7-0.4)=0.1$
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