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 first

The probability of choosing an advanced book is the number of advanced books divided by the total number of books. There are 3 advanced books and 11 total books. So the probability \(P(\text{advanced})\) is \(\frac{3}{11}\).

Step2: Calculate the probability of choosing a beginner book second

Since the book is replaced, the total number of books remains 11. There are 2 beginner books. So the probability \(P(\text{beginner})\) is \(\frac{2}{11}\).

Step3: Calculate the combined probability

For independent events (because the book is replaced), the combined probability is the product of the individual probabilities. So \(P = P(\text{advanced})\times P(\text{beginner})=\frac{3}{11}\times\frac{2}{11}\).

$$ \frac{3}{11}\times\frac{2}{11}=\frac{3\times2}{11\times11}=\frac{6}{121} $$

Answer:

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