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
1250⋅850
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction
1250⋅849
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
850
Start Fraction 8 over 50 End Fraction
1250
Responses
1250⋅850
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction
1250⋅849
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
850
Start Fraction 8 over 50 End Fraction
1250
Answers
GPT-4o mini
To find the probability of selecting a historical fiction book on the first selection and a romance novel on the second selection without replacement, we can follow these steps:
1. **Determine the total number of books on the shelf**: 50 books.
2. **Calculate the probability of selecting a historical fiction book first**:
- There are 12 historical fiction books.
- The probability of selecting a historical fiction book first is:
\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]
3. **Calculate the probability of selecting a romance novel second** after selecting a historical fiction book:
- After selecting one historical fiction book, there are now 49 books left (50 - 1).
- There are still 8 romance novels available.
- The probability of selecting a romance novel second is:
\[
P(\text{Romance second | Historical Fiction first}) = \frac{8}{49}
\]
4. **Multiply the probabilities**:
- The overall probability of both events occurring (selecting a historical fiction book first and then a romance novel) is:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first}) = \frac{12}{50} \times \frac{8}{49}
\]
5. **Calculate the final probability**:
\[
= \frac{12 \times 8}{50 \times 49} = \frac{96}{2450}
\]
This corresponds to the option:
\[
\frac{12}{50} \times \frac{8}{49}
\]
Thus, the correct response is:
**Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction**.
1. **Determine the total number of books on the shelf**: 50 books.
2. **Calculate the probability of selecting a historical fiction book first**:
- There are 12 historical fiction books.
- The probability of selecting a historical fiction book first is:
\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]
3. **Calculate the probability of selecting a romance novel second** after selecting a historical fiction book:
- After selecting one historical fiction book, there are now 49 books left (50 - 1).
- There are still 8 romance novels available.
- The probability of selecting a romance novel second is:
\[
P(\text{Romance second | Historical Fiction first}) = \frac{8}{49}
\]
4. **Multiply the probabilities**:
- The overall probability of both events occurring (selecting a historical fiction book first and then a romance novel) is:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first}) = \frac{12}{50} \times \frac{8}{49}
\]
5. **Calculate the final probability**:
\[
= \frac{12 \times 8}{50 \times 49} = \frac{96}{2450}
\]
This corresponds to the option:
\[
\frac{12}{50} \times \frac{8}{49}
\]
Thus, the correct response is:
**Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction**.