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Sagot :
To determine the probability that Nadia randomly picks up a reference book and then, without replacing it, picks up a nonfiction book, we will calculate it step by step.
1. Count the total number of books:
- Fiction books: [tex]\(10\)[/tex]
- Reference books: [tex]\(2\)[/tex]
- Nonfiction books: [tex]\(5\)[/tex]
So, the total number of books is:
[tex]\[ 10 + 2 + 5 = 17 \][/tex]
2. Calculate the probability of picking a reference book first:
The probability [tex]\(P(A)\)[/tex] of picking a reference book out of the 17 books is:
[tex]\[ P(A) = \frac{\text{Number of reference books}}{\text{Total number of books}} = \frac{2}{17} = 0.11764705882352941 \][/tex]
3. Calculate the probability of picking a nonfiction book second, given that a reference book was picked first:
After picking a reference book, there are [tex]\(16\)[/tex] books left:
[tex]\[ \text{Remaining total books} = 17 - 1 = 16 \][/tex]
Now there are still [tex]\(5\)[/tex] nonfiction books because we did not pick any nonfiction book initially.
The probability [tex]\(P(B|A)\)[/tex] of picking a nonfiction book from the remaining [tex]\(16\)[/tex] books is:
[tex]\[ P(B|A) = \frac{\text{Number of nonfiction books}}{\text{Remaining total books}} = \frac{5}{16} = 0.3125 \][/tex]
4. Compute the combined probability of both events occurring:
According to the rule of conditional probability, the combined probability [tex]\(P(A \text{ and } B)\)[/tex] of both events (picking a reference book first and then picking a nonfiction book without replacement) is:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B|A) = \left(\frac{2}{17}\right) \times \left(\frac{5}{16}\right) = 0.03676470588235294 \][/tex]
5. Convert the combined probability into fractional form to find the answer choice:
The calculated combined probability is [tex]\(0.03676470588235294\)[/tex], which in fractional form matches with:
[tex]\[ \frac{5}{136} \][/tex]
Hence, the probability that Nadia randomly picks up a reference book and then, without replacing it, picks up a nonfiction book is [tex]\(\boxed{\frac{5}{136}}\)[/tex].
1. Count the total number of books:
- Fiction books: [tex]\(10\)[/tex]
- Reference books: [tex]\(2\)[/tex]
- Nonfiction books: [tex]\(5\)[/tex]
So, the total number of books is:
[tex]\[ 10 + 2 + 5 = 17 \][/tex]
2. Calculate the probability of picking a reference book first:
The probability [tex]\(P(A)\)[/tex] of picking a reference book out of the 17 books is:
[tex]\[ P(A) = \frac{\text{Number of reference books}}{\text{Total number of books}} = \frac{2}{17} = 0.11764705882352941 \][/tex]
3. Calculate the probability of picking a nonfiction book second, given that a reference book was picked first:
After picking a reference book, there are [tex]\(16\)[/tex] books left:
[tex]\[ \text{Remaining total books} = 17 - 1 = 16 \][/tex]
Now there are still [tex]\(5\)[/tex] nonfiction books because we did not pick any nonfiction book initially.
The probability [tex]\(P(B|A)\)[/tex] of picking a nonfiction book from the remaining [tex]\(16\)[/tex] books is:
[tex]\[ P(B|A) = \frac{\text{Number of nonfiction books}}{\text{Remaining total books}} = \frac{5}{16} = 0.3125 \][/tex]
4. Compute the combined probability of both events occurring:
According to the rule of conditional probability, the combined probability [tex]\(P(A \text{ and } B)\)[/tex] of both events (picking a reference book first and then picking a nonfiction book without replacement) is:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B|A) = \left(\frac{2}{17}\right) \times \left(\frac{5}{16}\right) = 0.03676470588235294 \][/tex]
5. Convert the combined probability into fractional form to find the answer choice:
The calculated combined probability is [tex]\(0.03676470588235294\)[/tex], which in fractional form matches with:
[tex]\[ \frac{5}{136} \][/tex]
Hence, the probability that Nadia randomly picks up a reference book and then, without replacing it, picks up a nonfiction book is [tex]\(\boxed{\frac{5}{136}}\)[/tex].
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