To find the probability of selecting a historical fiction book first and then a romance novel second without replacement, we can apply the probabilities step-by-step.
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Probability of selecting a historical fiction book on the first selection: There are 12 historical fiction books out of 50 total books. So, the probability of selecting a historical fiction book first is:
\[ P(\text{Historical Fiction first}) = \frac{12}{50} \]
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Probability of selecting a romance novel on the second selection (after one historical fiction book has been removed): After removing one historical fiction book, there will be 49 books left. The number of romance novels remains at 8. Therefore, the probability of selecting a romance novel second is:
\[ P(\text{Romance second} | \text{Historical Fiction first}) = \frac{8}{49} \]
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Combined probability: To find the overall probability of both events happening (selecting a historical fiction book first and then a romance novel), we multiply the probabilities:
\[ P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second} | \text{Historical Fiction first}) = \frac{12}{50} \times \frac{8}{49} \]
So, the final answer simplifies to:
\[ \frac{12}{50} \cdot \frac{8}{49} \]
This matches option A. Therefore, the answer is:
A. \( \frac{12}{50} \cdot \frac{8}{49} \)