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.(1 point)
Responses
850
Start Fraction 8 over 50 End Fraction
1250
Start Fraction 12 over 50 End Fraction
1250⋅850
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction
1250⋅849
Responses
850
Start Fraction 8 over 50 End Fraction
1250
Start Fraction 12 over 50 End Fraction
1250⋅850
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction
1250⋅849
Answers
GPT-4o mini
To find the probability of selecting a historical fiction book first and a romance novel second without replacement, we need to calculate the probabilities step by step.
1. **Selecting the first book (historical fiction):**
- There are 12 works of historical fiction out of 50 total books.
- The probability of selecting a historical fiction book first is:
\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]
2. **Selecting the second book (romance):**
- After selecting one historical fiction book, there are now 49 books left on the shelf, including all 8 romance novels.
- The probability of selecting a romance novel second is:
\[
P(\text{Romance second} | \text{Historical Fiction first}) = \frac{8}{49}
\]
3. **Calculating the combined probability:**
- Multiply the probabilities of both events:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second} | \text{Historical Fiction first})
\]
\[
= \frac{12}{50} \times \frac{8}{49}
\]
Now, we can perform the multiplication to find the final probability:
\[
= \frac{12 \times 8}{50 \times 49} = \frac{96}{2450}
\]
Thus, the answer corresponds to the probability of selecting a historical fiction book first and a romance novel second.
From the options you've provided, it seems there is no exact match for \(\frac{96}{2450}\), but the closest answer could be interpreted as:
\[
\frac{12}{50} \cdot \frac{8}{49}
\]
This calculates to \(\frac{96}{2450}\), reinforcing our computation with correct values from the selections given.
Hence, the correct response to the question would be:
**1250⋅849** (as this equivalently suggests a similar structure to the calculation).
1. **Selecting the first book (historical fiction):**
- There are 12 works of historical fiction out of 50 total books.
- The probability of selecting a historical fiction book first is:
\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]
2. **Selecting the second book (romance):**
- After selecting one historical fiction book, there are now 49 books left on the shelf, including all 8 romance novels.
- The probability of selecting a romance novel second is:
\[
P(\text{Romance second} | \text{Historical Fiction first}) = \frac{8}{49}
\]
3. **Calculating the combined probability:**
- Multiply the probabilities of both events:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second} | \text{Historical Fiction first})
\]
\[
= \frac{12}{50} \times \frac{8}{49}
\]
Now, we can perform the multiplication to find the final probability:
\[
= \frac{12 \times 8}{50 \times 49} = \frac{96}{2450}
\]
Thus, the answer corresponds to the probability of selecting a historical fiction book first and a romance novel second.
From the options you've provided, it seems there is no exact match for \(\frac{96}{2450}\), but the closest answer could be interpreted as:
\[
\frac{12}{50} \cdot \frac{8}{49}
\]
This calculates to \(\frac{96}{2450}\), reinforcing our computation with correct values from the selections given.
Hence, the correct response to the question would be:
**1250⋅849** (as this equivalently suggests a similar structure to the calculation).