To find the probability that Federica selects a historical fiction book first and a romance novel second (without replacement), we can follow these steps:
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Determine the total number of books: There are 50 books in total.
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Find the probability of selecting a historical fiction book first: There are 12 historical fiction books. \[ P(\text{Historical Fiction first}) = \frac{12}{50} \]
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After selecting one book, find the number of books left: If she selects a historical fiction book first, there will be: \[ 50 - 1 = 49 \text{ books remaining} \] And the total number of romance novels remains at 8.
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Find the probability of selecting a romance novel second: \[ P(\text{Romance second | Historical Fiction first}) = \frac{8}{49} \]
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Calculate the combined probability: To find the probability of both events occurring (selecting a historical fiction book first and then a romance novel), you multiply the probabilities of both events: \[ P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first}) \] \[ = \frac{12}{50} \times \frac{8}{49} \]
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Calculate the final answer: \[ = \frac{12 \times 8}{50 \times 49} = \frac{96}{2450} \]
So, the probability of selecting a historical fiction book first and a romance novel second is \(\frac{96}{2450}\).
- Simplify if possible: We can divide the numerator and the denominator by 2: \[ \frac{96 \div 2}{2450 \div 2} = \frac{48}{1225} \]
Thus, the final probability is: \[ \frac{48}{1225} \]