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 nonfiction book, and then a mystery novel
Answers
GPT-4o mini
To find the probability of Federica selecting a nonfiction book first and then a mystery novel, we can follow these steps:
1. **Total number of books on the shelf**: 50
2. **Number of nonfiction books**: 15
3. **Number of mystery novels**: 15
Now we will calculate the probability step by step.
**Step 1: Probability of selecting a nonfiction book first.**
The probability of selecting a nonfiction book is given by the number of nonfiction books divided by the total number of books:
\[
P(\text{Nonfiction first}) = \frac{\text{Number of nonfiction books}}{\text{Total number of books}} = \frac{15}{50} = \frac{3}{10}
\]
**Step 2: Probability of selecting a mystery novel second (after one nonfiction book has been removed).**
If a nonfiction book is selected first, there will be 49 books left on the shelf (50 - 1 = 49). The number of mystery novels remains 15 since no mystery novel has been selected yet.
The probability of then selecting a mystery novel is:
\[
P(\text{Mystery novel second | Nonfiction first}) = \frac{\text{Number of mystery novels}}{\text{Total number of remaining books}} = \frac{15}{49}
\]
**Step 3: Combined probability of both events.**
The combined probability of both events (selecting a nonfiction book first and then a mystery novel) can be calculated by multiplying the probabilities of both steps:
\[
P(\text{Nonfiction first and then Mystery novel}) = P(\text{Nonfiction first}) \times P(\text{Mystery novel second | Nonfiction first}) = \frac{3}{10} \times \frac{15}{49}
\]
Calculating that gives:
\[
P = \frac{3 \times 15}{10 \times 49} = \frac{45}{490} = \frac{9}{98}
\]
Thus, the probability of selecting a nonfiction book first and then a mystery novel is \(\frac{9}{98}\).
1. **Total number of books on the shelf**: 50
2. **Number of nonfiction books**: 15
3. **Number of mystery novels**: 15
Now we will calculate the probability step by step.
**Step 1: Probability of selecting a nonfiction book first.**
The probability of selecting a nonfiction book is given by the number of nonfiction books divided by the total number of books:
\[
P(\text{Nonfiction first}) = \frac{\text{Number of nonfiction books}}{\text{Total number of books}} = \frac{15}{50} = \frac{3}{10}
\]
**Step 2: Probability of selecting a mystery novel second (after one nonfiction book has been removed).**
If a nonfiction book is selected first, there will be 49 books left on the shelf (50 - 1 = 49). The number of mystery novels remains 15 since no mystery novel has been selected yet.
The probability of then selecting a mystery novel is:
\[
P(\text{Mystery novel second | Nonfiction first}) = \frac{\text{Number of mystery novels}}{\text{Total number of remaining books}} = \frac{15}{49}
\]
**Step 3: Combined probability of both events.**
The combined probability of both events (selecting a nonfiction book first and then a mystery novel) can be calculated by multiplying the probabilities of both steps:
\[
P(\text{Nonfiction first and then Mystery novel}) = P(\text{Nonfiction first}) \times P(\text{Mystery novel second | Nonfiction first}) = \frac{3}{10} \times \frac{15}{49}
\]
Calculating that gives:
\[
P = \frac{3 \times 15}{10 \times 49} = \frac{45}{490} = \frac{9}{98}
\]
Thus, the probability of selecting a nonfiction book first and then a mystery novel is \(\frac{9}{98}\).