To find the probability of selecting a historical fiction book first and then a romance novel second, we need to consider the total number of books and the respective counts for the categories of interest.
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Determine Total Books: There are a total of 50 books on the shelf.
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Successful Outcomes:
- First, we need to determine the number of historical fiction books, which is 12.
- Next, if a historical fiction book is selected first, the total number of books left would be 49 (because we're selecting without replacement).
- The number of romance novels available remains the same for the second selection, which is 8.
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Calculate the Probability for Each Selection:
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The probability of selecting a historical fiction book first is: \[ P(\text{Historical Fiction first}) = \frac{12}{50} \]
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After selecting a historical fiction book, the probability of selecting a romance novel second is: \[ P(\text{Romance second} | \text{Historical Fiction first}) = \frac{8}{49} \]
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Calculate the Joint Probability:
- The total probability of both events happening in sequence is the product of the individual probabilities: \[ P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second} | \text{Historical Fiction first}) \] \[ P(\text{Historical Fiction first and Romance second}) = \frac{12}{50} \times \frac{8}{49} \]
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Simplify:
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Calculate the product: \[ P = \frac{12 \times 8}{50 \times 49} = \frac{96}{2450} \]
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Simplifying the fraction: \[ \frac{96}{2450} = \frac{48}{1225} \quad \text{(since both 96 and 2450 can be divided by 2)} \]
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Final Probability: The probability of selecting a historical fiction book first and a romance novel second is: \[ \boxed{\frac{48}{1225}} \]