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. Ravi must choose 2 books and doesn't really care which kind they are. He decides to randomly pick 1 and then pick another without replacement. If Federica randomly selects 2 books from the shelf, what is the probability of selecting a nonfiction book, and then a mystery novel
Answers
GPT-4o mini
To find the probability that Federica selects a nonfiction book first and then a mystery novel, we can follow these steps:
1. **Calculate the total number of books**: There are 50 books on the shelf.
2. **Probability of selecting a nonfiction book first**:
- There are 15 nonfiction books.
- The probability of selecting a nonfiction book first is:
\[
P(\text{Nonfiction first}) = \frac{15}{50} = \frac{3}{10}
\]
3. **Determine the number of books left after selecting a nonfiction book**:
- After selecting one nonfiction book, there will be 49 books left on the shelf (50 - 1 = 49).
4. **Calculate the number of mystery novels remaining**:
- There are still 15 mystery novels on the shelf because we didn’t select any mystery novel yet.
5. **Probability of selecting a mystery novel second**:
- The probability of selecting a mystery novel after having selected a nonfiction book is:
\[
P(\text{Mystery second | Nonfiction first}) = \frac{15}{49}
\]
6. **Combine the probabilities**:
- The overall probability of selecting a nonfiction book first and then a mystery novel is the product of the individual probabilities:
\[
P(\text{Nonfiction first and Mystery second}) = P(\text{Nonfiction first}) \times P(\text{Mystery second | Nonfiction first}) = \frac{3}{10} \times \frac{15}{49}
\]
7. **Calculate the final probability**:
\[
P(\text{Nonfiction first and Mystery second}) = \frac{3 \times 15}{10 \times 49} = \frac{45}{490} = \frac{9}{98}
\]
Thus, the probability of selecting a nonfiction book first and then a mystery novel is:
\[
\frac{9}{98}
\]
1. **Calculate the total number of books**: There are 50 books on the shelf.
2. **Probability of selecting a nonfiction book first**:
- There are 15 nonfiction books.
- The probability of selecting a nonfiction book first is:
\[
P(\text{Nonfiction first}) = \frac{15}{50} = \frac{3}{10}
\]
3. **Determine the number of books left after selecting a nonfiction book**:
- After selecting one nonfiction book, there will be 49 books left on the shelf (50 - 1 = 49).
4. **Calculate the number of mystery novels remaining**:
- There are still 15 mystery novels on the shelf because we didn’t select any mystery novel yet.
5. **Probability of selecting a mystery novel second**:
- The probability of selecting a mystery novel after having selected a nonfiction book is:
\[
P(\text{Mystery second | Nonfiction first}) = \frac{15}{49}
\]
6. **Combine the probabilities**:
- The overall probability of selecting a nonfiction book first and then a mystery novel is the product of the individual probabilities:
\[
P(\text{Nonfiction first and Mystery second}) = P(\text{Nonfiction first}) \times P(\text{Mystery second | Nonfiction first}) = \frac{3}{10} \times \frac{15}{49}
\]
7. **Calculate the final probability**:
\[
P(\text{Nonfiction first and Mystery second}) = \frac{3 \times 15}{10 \times 49} = \frac{45}{490} = \frac{9}{98}
\]
Thus, the probability of selecting a nonfiction book first and then a mystery novel is:
\[
\frac{9}{98}
\]