Question

There are 50 books on a shelf. They include 15 mystery novels, 12 works of historical fiction, 8 romance novels, and 15 nonfiction books. If Federica randomly selects 2 books from the shelf, what is the probability of selecting a historical fiction book on the first selection and selecting romance on the second selection? Assume the books are taken without replacement.

Answer 1

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.

1. **Determine Total Books**: There are a total of 50 books on the shelf.

2. **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.

3. **Calculate the Probability for Each Selection**:
- The probability of selecting a historical fiction book first is:
\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]

- After selecting a historical fiction book, the probability of selecting a romance novel second is:
\[
P(\text{Romance second} | \text{Historical Fiction first}) = \frac{8}{49}
\]

4. **Calculate the Joint Probability**:
- The total probability of both events happening in sequence is the product of the individual probabilities:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second} | \text{Historical Fiction first})
\]
\[
P(\text{Historical Fiction first and Romance second}) = \frac{12}{50} \times \frac{8}{49}
\]

5. **Simplify**:
- Calculate the product:
\[
P = \frac{12 \times 8}{50 \times 49} = \frac{96}{2450}
\]

- Simplifying the fraction:
\[
\frac{96}{2450} = \frac{48}{1225} \quad \text{(since both 96 and 2450 can be divided by 2)}
\]

6. **Final Probability**:
The probability of selecting a historical fiction book first and a romance novel second is:
\[
\boxed{\frac{48}{1225}}
\]