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.
8/50
12/50*8/49
12/50*8/50
12/50
8/50
12/50*8/49
12/50*8/50
12/50
Answers
GPT-4o mini
To find the probability of Ravi selecting a nonfiction book first and then a mystery novel second (without replacement), we can break the problem down into two steps:
1. **Probability of selecting a nonfiction book first:**
There are 15 nonfiction books out of a total of 50 books.
\[
P(\text{Nonfiction first}) = \frac{15}{50}
\]
2. **Probability of selecting a mystery novel second after selecting a nonfiction book:**
After Ravi selects a nonfiction book, there will be 49 books left on the shelf, and 15 of those will be mystery novels.
\[
P(\text{Mystery novel second | Nonfiction first}) = \frac{15}{49}
\]
Now, we can multiply these two probabilities to find the overall probability of selecting a nonfiction book first and a mystery novel second:
\[
P(\text{Nonfiction first and then Mystery}) = P(\text{Nonfiction first}) \times P(\text{Mystery novel second | Nonfiction first}) = \frac{15}{50} \times \frac{15}{49}
\]
Calculating this gives:
\[
\frac{15}{50} \times \frac{15}{49} = \frac{15 \times 15}{50 \times 49} = \frac{225}{2450}
\]
Now we want to simplify that:
\[
\frac{225}{2450} = \frac{9}{98}
\]
So the final probability of selecting a nonfiction book first and then a mystery novel is \(\frac{9}{98}\).
Now, looking at the options you provided:
- 8/50
- \( \frac{12}{50} \times \frac{8}{49} \)
- \( \frac{12}{50} \times \frac{8}{50} \)
- \( \frac{12}{50} \)
None of these options represent the correct conditional probabilities for selecting a nonfiction book first and then a mystery novel. The probability we calculated does not match any of the options directly. However, the calculation of the steps involved demonstrates the correct method used to arrive at the probability.
Thus, the answer to your specific question is that none of the listed options is correct for the probability of selecting a nonfiction book first and then a mystery novel second. The correct probability is \(\frac{9}{98}\).
1. **Probability of selecting a nonfiction book first:**
There are 15 nonfiction books out of a total of 50 books.
\[
P(\text{Nonfiction first}) = \frac{15}{50}
\]
2. **Probability of selecting a mystery novel second after selecting a nonfiction book:**
After Ravi selects a nonfiction book, there will be 49 books left on the shelf, and 15 of those will be mystery novels.
\[
P(\text{Mystery novel second | Nonfiction first}) = \frac{15}{49}
\]
Now, we can multiply these two probabilities to find the overall probability of selecting a nonfiction book first and a mystery novel second:
\[
P(\text{Nonfiction first and then Mystery}) = P(\text{Nonfiction first}) \times P(\text{Mystery novel second | Nonfiction first}) = \frac{15}{50} \times \frac{15}{49}
\]
Calculating this gives:
\[
\frac{15}{50} \times \frac{15}{49} = \frac{15 \times 15}{50 \times 49} = \frac{225}{2450}
\]
Now we want to simplify that:
\[
\frac{225}{2450} = \frac{9}{98}
\]
So the final probability of selecting a nonfiction book first and then a mystery novel is \(\frac{9}{98}\).
Now, looking at the options you provided:
- 8/50
- \( \frac{12}{50} \times \frac{8}{49} \)
- \( \frac{12}{50} \times \frac{8}{50} \)
- \( \frac{12}{50} \)
None of these options represent the correct conditional probabilities for selecting a nonfiction book first and then a mystery novel. The probability we calculated does not match any of the options directly. However, the calculation of the steps involved demonstrates the correct method used to arrive at the probability.
Thus, the answer to your specific question is that none of the listed options is correct for the probability of selecting a nonfiction book first and then a mystery novel second. The correct probability is \(\frac{9}{98}\).