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
Answers
GPT-4o mini
To find the probability of selecting a historical fiction book first and then a romance novel second without replacement, we can follow these steps.
1. **Calculate the total number of books**:
There are 50 books in total.
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**:
If one historical fiction book has already been taken, there are now 49 books left.
The number of romance novels remains 8.
The probability of selecting a romance novel second is:
\[
P(\text{Romance second | Historical Fiction first}) = \frac{8}{49}
\]
4. **Combine the probabilities**:
The combined probability of both events happening (selecting a historical fiction book first and then a romance novel) is found by multiplying the two probabilities:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first}) = \frac{12}{50} \times \frac{8}{49}
\]
5. **Calculate the result**:
\[
P(\text{Historical Fiction first and Romance second}) = \frac{12 \times 8}{50 \times 49}
\]
\[
= \frac{96}{2450}
\]
6. **Simplify the probability**:
To simplify \( \frac{96}{2450} \):
- The greatest common divisor (GCD) of 96 and 2450 is 2.
- Now divide both the numerator and the denominator by 2:
\[
= \frac{48}{1225}
\]
Thus, the probability of selecting a historical fiction book first and a romance novel second is:
\[
\boxed{\frac{48}{1225}}
\]
1. **Calculate the total number of books**:
There are 50 books in total.
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**:
If one historical fiction book has already been taken, there are now 49 books left.
The number of romance novels remains 8.
The probability of selecting a romance novel second is:
\[
P(\text{Romance second | Historical Fiction first}) = \frac{8}{49}
\]
4. **Combine the probabilities**:
The combined probability of both events happening (selecting a historical fiction book first and then a romance novel) is found by multiplying the two probabilities:
\[
P(\text{Historical Fiction first and Romance second}) = P(\text{Historical Fiction first}) \times P(\text{Romance second | Historical Fiction first}) = \frac{12}{50} \times \frac{8}{49}
\]
5. **Calculate the result**:
\[
P(\text{Historical Fiction first and Romance second}) = \frac{12 \times 8}{50 \times 49}
\]
\[
= \frac{96}{2450}
\]
6. **Simplify the probability**:
To simplify \( \frac{96}{2450} \):
- The greatest common divisor (GCD) of 96 and 2450 is 2.
- Now divide both the numerator and the denominator by 2:
\[
= \frac{48}{1225}
\]
Thus, the probability of selecting a historical fiction book first and a romance novel second is:
\[
\boxed{\frac{48}{1225}}
\]