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
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
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
850
Responses
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
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
850
Answers
GPT-4o mini
To find the probability of selecting a historical fiction book first and a romance novel second without replacement, we can follow these steps:
1. **Calculate the probability of selecting a historical fiction book first.**
There are 12 historical fiction books out of 50 total books.
\[
P(\text{H.F. first}) = \frac{12}{50}
\]
2. **Calculate the probability of selecting a romance book second after selecting a historical fiction book.**
After one historical fiction book is removed, there are now 49 books left in total, and still 8 romance novels.
\[
P(\text{Romance second | H.F. first}) = \frac{8}{49}
\]
3. **Multiply the two probabilities together to get the total probability of both events happening in sequence.**
\[
P(\text{H.F. first and Romance second}) = P(\text{H.F. first}) \times P(\text{Romance second | H.F. first}) = \frac{12}{50} \times \frac{8}{49}
\]
Calculating this gives:
\[
P(\text{H.F. first and Romance second}) = \frac{12}{50} \times \frac{8}{49} = \frac{96}{2450}
\]
Now, we can check the given responses to match this value:
- Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction is incorrect because the second fraction should have a denominator of 49, not 50.
- Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction is the correct representation of our calculation.
So the correct answer is:
**Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction** (which represents the probability).
1. **Calculate the probability of selecting a historical fiction book first.**
There are 12 historical fiction books out of 50 total books.
\[
P(\text{H.F. first}) = \frac{12}{50}
\]
2. **Calculate the probability of selecting a romance book second after selecting a historical fiction book.**
After one historical fiction book is removed, there are now 49 books left in total, and still 8 romance novels.
\[
P(\text{Romance second | H.F. first}) = \frac{8}{49}
\]
3. **Multiply the two probabilities together to get the total probability of both events happening in sequence.**
\[
P(\text{H.F. first and Romance second}) = P(\text{H.F. first}) \times P(\text{Romance second | H.F. first}) = \frac{12}{50} \times \frac{8}{49}
\]
Calculating this gives:
\[
P(\text{H.F. first and Romance second}) = \frac{12}{50} \times \frac{8}{49} = \frac{96}{2450}
\]
Now, we can check the given responses to match this value:
- Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction is incorrect because the second fraction should have a denominator of 49, not 50.
- Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction is the correct representation of our calculation.
So the correct answer is:
**Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction** (which represents the probability).