To solve the problem of finding the probability that both Audrey and Ryan pick nonfiction books, let's break it down into smaller steps.
1. Identify the given numbers:
- Fiction books: 180
- Nonfiction books: 400
- Total books: [tex]\(180 + 400 = 580\)[/tex]
2. Calculate the probability that Audrey picks a nonfiction book first:
- Probability (Audrey picks nonfiction) [tex]\(= \frac{\text{Number of nonfiction books}}{\text{Total number of books}} = \frac{400}{580} = \frac{20}{29}\)[/tex]
3. Calculate the probability that Ryan picks a nonfiction book after Audrey has picked one:
- After Audrey picks a nonfiction book, the number of nonfiction books decreases by one, so there are now [tex]\(400 - 1 = 399\)[/tex] nonfiction books left.
- The total number of books now is [tex]\(580 - 1 = 579\)[/tex].
- Probability (Ryan picks nonfiction given Audrey already picked nonfiction) [tex]\(= \frac{399}{579} = \frac{21}{29}\)[/tex]
4. Determine the probability that both Audrey and Ryan pick nonfiction books:
- The combined probability of both events happening is the product of these individual probabilities:
- Probability (both pick nonfiction) [tex]\(= \frac{20}{29} \times \frac{21}{29} = \frac{420}{841}\)[/tex].
However, this exact fraction (based on the combined probabilities) needs to be evaluated against the given options. Sometimes in questions, there might be a slight ambiguity or error, so let's look at possible closest matching options provided:
Given the options:
A) [tex]\(\frac{180 \times 189}{400 \times 400}\)[/tex]
B) [tex]\(\frac{180 \times 179}{400 \times 399}\)[/tex]
C) [tex]\(\frac{180 \times 179}{400 \times 400}\)[/tex]
D) [tex]\(\frac{180 \times 189}{400 \times 399}\)[/tex]
- Here, since 180 was mistakenly taken instead of 400 which causes a mismatch, none of the given answers precisely match the required probability.
However, if any cause of confusion in options, cross-verification against calculated correct probability can help in identifying inaccuracies in answer options given.
Hence for checking valid closest nearest accuracy probability overall combined:
The correct answer according to the problem's setup should reflect in proper correct precise fractions matching probabilities for correctly solved steps in solution evaluations.
