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Nico owns 11 instructional piano books: two beginner books, six intermediate books, and three advanced books.

If two books are chosen randomly from the collection, one at a time, and replaced after each pick, what is the probability that he first chooses an advanced book and then a beginner book?

A. [tex]\(\frac{5}{121}\)[/tex]
B. [tex]\(\frac{6}{121}\)[/tex]
C. [tex]\(\frac{5}{11}\)[/tex]
D. [tex]\(\frac{6}{11}\)[/tex]

Sagot :

GinnyAnswer
To find the probability that Nico first chooses an advanced book and then a beginner book with replacement, we can follow these detailed steps:

1. Total Number of Books:
Nico has 11 books in total.

2. Advanced Books:
Nico has 3 advanced books out of the 11.

3. Beginner Books:
Nico has 2 beginner books out of the 11.

4. Probability of Picking an Advanced Book First:
The probability of picking an advanced book in the first draw is found by dividing the number of advanced books by the total number of books:
[tex]\[ \text{Probability of advanced book first} = \frac{3}{11} \][/tex]

5. Probability of Picking a Beginner Book Second:
Since the books are replaced after each pick, the total number of books remains the same for the second draw. The probability of picking a beginner book in the second draw is:
[tex]\[ \text{Probability of beginner book second} = \frac{2}{11} \][/tex]

6. Combined Probability:
The combined probability of both events happening (picking an advanced book first and then picking a beginner book) is the product of their individual probabilities:
[tex]\[ \text{Combined probability} = \left( \frac{3}{11} \right) \times \left( \frac{2}{11} \right) = \frac{3 \times 2}{11 \times 11} = \frac{6}{121} \][/tex]

Thus, the probability that Nico picks an advanced book first and a beginner book second with replacement is:
[tex]\[ \boxed{\frac{6}{121}} \][/tex]
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