To find the probability of selecting a historical fiction book first and a romance novel second without replacement, we can follow these steps:
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Calculate the probability of selecting a historical fiction book first.
There are 12 historical fiction books out of a total of 50 books. Therefore, the probability of selecting a historical fiction book first is:\[ P(\text{Historical Fiction first}) = \frac{12}{50} \]
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Calculate the probability of selecting a romance novel second after having selected a historical fiction book first.
After selecting the historical fiction book, there are now 49 books left on the shelf, and still 8 romance novels. Therefore, the probability of selecting a romance novel second is:\[ P(\text{Romance second | Historical Fiction first}) = \frac{8}{49} \]
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Calculate the combined probability of both events occurring.
The total probability of both selecting a historical fiction book first and then a romance novel second is the product of the individual probabilities:\[ P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first}) \]
\[ = \frac{12}{50} \times \frac{8}{49} \]
\[ = \frac{12 \times 8}{50 \times 49} \]
\[ = \frac{96}{2450} \]
Looking at the options provided, the closest match to our calculations can be represented as:
\[ \frac{12}{50} \times \frac{8}{49} \]
This corresponds to the option:
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction.