Question

nico owns 11 instructional piano books. two are beginner books, six are intermediate books, and three are advanced books. if two books are randomly chosen 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 chooses a beginner book?

$\frac{5}{121}$

$\frac{6}{121}$

$\frac{5}{11}$

$\frac{6}{11}$

Explanation:

Step1: Calculate the probability of choosing an advanced book

The total number of books is \(n = 11\). The number of advanced books is \(m_1=3\). The probability of choosing an advanced book \(P_1=\frac{3}{11}\) (using the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)).

Step2: Calculate the probability of choosing a beginner book

Since the book is replaced, the total number of books is still \(n = 11\). The number of beginner books is \(m_2 = 2\). The probability of choosing a beginner book \(P_2=\frac{2}{11}\).

Step3: Calculate the combined probability

For independent events (because the book is replaced), the combined probability \(P = P_1\times P_2\). Substitute \(P_1=\frac{3}{11}\) and \(P_2=\frac{2}{11}\) into the formula: \(P=\frac{3}{11}\times\frac{2}{11}=\frac{6}{121}\).

Answer:

\(\frac{6}{121}\) (the second option)