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
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
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
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
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 a total of 50 books. Therefore, the probability of selecting a historical fiction book first is:
\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]
2. **Calculate the probability of selecting a romance novel second after having selected a historical fiction book first.**
After selecting the historical fiction book, there are now 49 books left on the shelf, and still 8 romance novels. Therefore, the probability of selecting a romance novel second is:
\[
P(\text{Romance second | Historical Fiction first}) = \frac{8}{49}
\]
3. **Calculate the combined probability of both events occurring.**
The total probability of both selecting a historical fiction book first and then a romance novel second is the product of the individual probabilities:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first})
\]
\[
= \frac{12}{50} \times \frac{8}{49}
\]
\[
= \frac{12 \times 8}{50 \times 49}
\]
\[
= \frac{96}{2450}
\]
Looking at the options provided, the closest match to our calculations can be represented as:
\[
\frac{12}{50} \times \frac{8}{49}
\]
This corresponds to the option:
**Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction.**
1. **Calculate the probability of selecting a historical fiction book first.**
There are 12 historical fiction books out of a total of 50 books. Therefore, the probability of selecting a historical fiction book first is:
\[
P(\text{Historical Fiction first}) = \frac{12}{50}
\]
2. **Calculate the probability of selecting a romance novel second after having selected a historical fiction book first.**
After selecting the historical fiction book, there are now 49 books left on the shelf, and still 8 romance novels. Therefore, the probability of selecting a romance novel second is:
\[
P(\text{Romance second | Historical Fiction first}) = \frac{8}{49}
\]
3. **Calculate the combined probability of both events occurring.**
The total probability of both selecting a historical fiction book first and then a romance novel second is the product of the individual probabilities:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first})
\]
\[
= \frac{12}{50} \times \frac{8}{49}
\]
\[
= \frac{12 \times 8}{50 \times 49}
\]
\[
= \frac{96}{2450}
\]
Looking at the options provided, the closest match to our calculations can be represented as:
\[
\frac{12}{50} \times \frac{8}{49}
\]
This corresponds to the option:
**Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction.**