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Sagot :
Let's break down the problem step by step.
1. Determine the number of fiction and nonfiction books:
- Let [tex]\( N \)[/tex] represent the number of nonfiction books.
- The number of fiction books will be [tex]\( N + 40 \)[/tex].
- The total number of books was given as 400:
[tex]\[ \text{Total books} = \text{Fiction books} + \text{Nonfiction books} \implies 400 = (N + 40) + N \][/tex]
- Solving for [tex]\( N \)[/tex]:
[tex]\[ 400 = 2N + 40 \implies 360 = 2N \implies N = 180 \][/tex]
- There are 180 nonfiction books and [tex]\( 180 + 40 = 220 \)[/tex] fiction books.
2. Probability that Audrey picks a nonfiction book first:
- The total number of books at the beginning is 400.
- The number of nonfiction books currently is 180.
- Therefore, the probability that Audrey picks a nonfiction book is:
[tex]\[ P(\text{Audrey picks nonfiction}) = \frac{\text{Number of Nonfiction Books}}{\text{Total Number of Books}} = \frac{180}{400} = 0.45 \][/tex]
3. Probability that Ryan picks a nonfiction book second:
- After Audrey has picked a book, there are 399 books left in total.
- If Audrey has picked a nonfiction book, there are now 179 nonfiction books remaining.
- Therefore, the probability that Ryan picks one of these remaining nonfiction books is:
[tex]\[ P(\text{Ryan picks nonfiction} \mid \text{Audrey picked nonfiction}) = \frac{\text{Remaining Nonfiction Books}}{\text{Remaining Total Books}} = \frac{179}{399} \approx 0.4486 \][/tex]
4. Combined probability that both Audrey and Ryan pick nonfiction books:
- The combined probability is the product of the individual probabilities:
[tex]\[ P(\text{Both pick nonfiction}) = P(\text{Audrey picks nonfiction}) \times P(\text{Ryan picks nonfiction} \mid \text{Audrey picked nonfiction}) = 0.45 \times 0.4486 \approx 0.2019 \][/tex]
Now, expressing this in terms of fractions and checking the given multiple-choice options, it becomes:
[tex]\[ P(\text{Both pick nonfiction}) = \frac{180}{400} \times \frac{179}{399} \][/tex]
This matches option B:
[tex]\[ \boxed{\frac{180 \times 179}{400 \times 399}} \][/tex]
1. Determine the number of fiction and nonfiction books:
- Let [tex]\( N \)[/tex] represent the number of nonfiction books.
- The number of fiction books will be [tex]\( N + 40 \)[/tex].
- The total number of books was given as 400:
[tex]\[ \text{Total books} = \text{Fiction books} + \text{Nonfiction books} \implies 400 = (N + 40) + N \][/tex]
- Solving for [tex]\( N \)[/tex]:
[tex]\[ 400 = 2N + 40 \implies 360 = 2N \implies N = 180 \][/tex]
- There are 180 nonfiction books and [tex]\( 180 + 40 = 220 \)[/tex] fiction books.
2. Probability that Audrey picks a nonfiction book first:
- The total number of books at the beginning is 400.
- The number of nonfiction books currently is 180.
- Therefore, the probability that Audrey picks a nonfiction book is:
[tex]\[ P(\text{Audrey picks nonfiction}) = \frac{\text{Number of Nonfiction Books}}{\text{Total Number of Books}} = \frac{180}{400} = 0.45 \][/tex]
3. Probability that Ryan picks a nonfiction book second:
- After Audrey has picked a book, there are 399 books left in total.
- If Audrey has picked a nonfiction book, there are now 179 nonfiction books remaining.
- Therefore, the probability that Ryan picks one of these remaining nonfiction books is:
[tex]\[ P(\text{Ryan picks nonfiction} \mid \text{Audrey picked nonfiction}) = \frac{\text{Remaining Nonfiction Books}}{\text{Remaining Total Books}} = \frac{179}{399} \approx 0.4486 \][/tex]
4. Combined probability that both Audrey and Ryan pick nonfiction books:
- The combined probability is the product of the individual probabilities:
[tex]\[ P(\text{Both pick nonfiction}) = P(\text{Audrey picks nonfiction}) \times P(\text{Ryan picks nonfiction} \mid \text{Audrey picked nonfiction}) = 0.45 \times 0.4486 \approx 0.2019 \][/tex]
Now, expressing this in terms of fractions and checking the given multiple-choice options, it becomes:
[tex]\[ P(\text{Both pick nonfiction}) = \frac{180}{400} \times \frac{179}{399} \][/tex]
This matches option B:
[tex]\[ \boxed{\frac{180 \times 179}{400 \times 399}} \][/tex]
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