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⋅849
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
1250
Start Fraction 12 over 50 End Fraction
850
Start Fraction 8 over 50 End Fraction
1250⋅850
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction
Responses
1250⋅849
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 49 End Fraction
1250
Start Fraction 12 over 50 End Fraction
850
Start Fraction 8 over 50 End Fraction
1250⋅850
Start Fraction 12 over 50 End Fraction times Start Fraction 8 over 50 End Fraction
Answers
GPT-4o mini
To find the probability of selecting a historical fiction book first and a romance novel second, we can break down the problem step by step.
1. **Probability of selecting a historical fiction book first**:
- There are 12 historical fiction books out of a total of 50 books.
- The probability \( P(HF) \) is given by:
\[
P(HF) = \frac{12}{50}
\]
2. **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 on the shelf.
- There are still 8 romance novels left.
- The probability \( P(R | HF) \) of selecting a romance novel after already having selected a historical fiction book is given by:
\[
P(R | HF) = \frac{8}{49}
\]
3. **Combine the probabilities**:
- The total probability \( P(HF \text{ and } R) \) is calculated using the multiplication rule for independent events:
\[
P(HF \text{ and } R) = P(HF) \times P(R | HF)
\]
Plugging in the probabilities:
\[
P(HF \text{ and } R) = \frac{12}{50} \times \frac{8}{49}
\]
4. **Final Probability Calculation**:
\[
P(HF \text{ and } R) = \frac{12 \times 8}{50 \times 49} = \frac{96}{2450}
\]
Thus, the final answer for the probability of selecting a historical fiction book first and a romance book second is:
\[
\frac{12}{50} \times \frac{8}{49}
\]
From your provided responses, the correct expression for the probability is:
\[
\frac{12}{50} \times \frac{8}{49}
\]
1. **Probability of selecting a historical fiction book first**:
- There are 12 historical fiction books out of a total of 50 books.
- The probability \( P(HF) \) is given by:
\[
P(HF) = \frac{12}{50}
\]
2. **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 on the shelf.
- There are still 8 romance novels left.
- The probability \( P(R | HF) \) of selecting a romance novel after already having selected a historical fiction book is given by:
\[
P(R | HF) = \frac{8}{49}
\]
3. **Combine the probabilities**:
- The total probability \( P(HF \text{ and } R) \) is calculated using the multiplication rule for independent events:
\[
P(HF \text{ and } R) = P(HF) \times P(R | HF)
\]
Plugging in the probabilities:
\[
P(HF \text{ and } R) = \frac{12}{50} \times \frac{8}{49}
\]
4. **Final Probability Calculation**:
\[
P(HF \text{ and } R) = \frac{12 \times 8}{50 \times 49} = \frac{96}{2450}
\]
Thus, the final answer for the probability of selecting a historical fiction book first and a romance book second is:
\[
\frac{12}{50} \times \frac{8}{49}
\]
From your provided responses, the correct expression for the probability is:
\[
\frac{12}{50} \times \frac{8}{49}
\]