A public library has a fiction section and a nonfiction section. The fiction section has 8 shelves, and each shelf...

1. TRANSLATE the problem information

• Given information:
  • Fiction section: 8 shelves, each with 10 hardcover + 15 paperback books
  • Nonfiction section: 6 shelves, each with 10 hardcover + 10 paperback books
  • Need probability of fiction book GIVEN it's hardcover
• The phrase "given that the book is a hardcover" signals conditional probability

2. INFER the approach

• This asks for \(\mathrm{P(Fiction\, |\, Hardcover)}\), which means: "Among all hardcover books, what fraction are fiction?"
• We need to count hardcover fiction books and total hardcover books
• Use conditional probability concept: focus only on the "given" category (hardcover books)

3. TRANSLATE and calculate hardcover books in each section

• Fiction hardcover books: \(\mathrm{8\, shelves \times 10\, books/shelf = 80\, books}\)
• Nonfiction hardcover books: \(\mathrm{6\, shelves \times 10\, books/shelf = 60\, books}\)
• Total hardcover books: \(\mathrm{80 + 60 = 140\, books}\)

4. INFER the probability calculation

• Among the 140 hardcover books, 80 are fiction
• \(\mathrm{P(Fiction\, |\, Hardcover) = \frac{80}{140}}\)

5. SIMPLIFY the fraction

• Find GCD of 80 and 140: GCD = 20
• \(\mathrm{80 \div 20 = 4}\), and \(\mathrm{140 \div 20 = 7}\)
• Final answer: \(\mathrm{\frac{4}{7}}\)

Answer: D) \(\mathrm{\frac{4}{7}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread "given that it's hardcover" and calculate \(\mathrm{P(Hardcover\, |\, Fiction)}\) instead of \(\mathrm{P(Fiction\, |\, Hardcover)}\).

They might calculate: \(\mathrm{\frac{80\, fiction\, hardcover\, books}{200\, total\, fiction\, books} = \frac{80}{200} = \frac{2}{5}}\)

This leads them to select Choice B (\(\mathrm{\frac{2}{5}}\)).

Second Most Common Error:

Missing conditional probability concept: Students calculate the probability of randomly selecting a fiction hardcover book from ALL books in the library.

They calculate: \(\mathrm{\frac{80\, fiction\, hardcover}{320\, total\, books} = \frac{80}{320} = \frac{1}{4}}\)

This may lead them to select Choice A (\(\mathrm{\frac{1}{4}}\)).

The Bottom Line:

The key challenge is recognizing that "given that" restricts our sample space to only hardcover books, not all books in the library. Students must understand they're finding what fraction of hardcover books are fiction, not what fraction of all books are fiction hardcover.