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.

A. 12/50 ⋅ 8/49
B. 12/50 ⋅ 8/50
C. 12/50
D. 8/50

Answer 1

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.

1. **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:

\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]

2. **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:

\[
P(\text{Romance second} | \text{Historical Fiction first}) = \frac{8}{49}
\]

3. **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. \( \frac{12}{50} \cdot \frac{8}{49} \)**