To solve this problem step-by-step, let's break it down into manageable parts.
1. Total Number of Books:
Nico has a total of 11 instructional piano books.
2. Category of Books:
- Advanced books: 3
- Beginner books: 2
- Intermediate books: 6
3. First Pick - Probability of Choosing an Advanced Book:
The probability of selecting an advanced book on the first pick is calculated by dividing the number of advanced books by the total number of books:
[tex]\[
P(\text{Advanced Book First}) = \frac{\text{Number of Advanced Books}}{\text{Total Number of Books}} = \frac{3}{11}
\][/tex]
4. Second Pick - Probability of Choosing a Beginner Book:
Since the book is replaced after each pick, the total number of books remains the same. The probability of selecting a beginner book on the second pick is:
[tex]\[
P(\text{Beginner Book Second}) = \frac{\text{Number of Beginner Books}}{\text{Total Number of Books}} = \frac{2}{11}
\][/tex]
5. Combined Probability:
Since we are looking for the probability of two independent events happening consecutively (choosing an advanced book first and a beginner book second), we multiply the probabilities of these two events:
[tex]\[
P(\text{Advanced then Beginner}) = P(\text{Advanced Book First}) \times P(\text{Beginner Book Second})
\][/tex]
Substituting the values we found:
[tex]\[
P(\text{Advanced then Beginner}) = \left(\frac{3}{11}\right) \times \left(\frac{2}{11}\right)
\][/tex]
[tex]\[
P(\text{Advanced then Beginner}) = \frac{3 \times 2}{11 \times 11} = \frac{6}{121}
\][/tex]
Therefore, the probability that Nico first chooses an advanced book and then chooses a beginner book is:
[tex]\[
\boxed{\frac{6}{121}}
\][/tex]
