There are n nonfiction books and 12 fiction books on a bookshelf. If one of these books is selected at...

1. TRANSLATE the problem information

• Given information:
  • n nonfiction books on the shelf
  • 12 fiction books on the shelf
  • One book is selected at random
  • Need to find probability of selecting a nonfiction book

2. INFER the approach

• This is a basic probability problem
• We need to use: Probability = \(\frac{\text{favorable outcomes}}{\text{total outcomes}}\)
• Favorable outcomes = selecting nonfiction book
• Total outcomes = all possible book selections

3. TRANSLATE to set up the probability fraction

• Number of favorable outcomes (nonfiction books) = n
• Total number of outcomes (all books) = n + 12
• Probability = \(\frac{n}{n+12}\)

Answer: B. \(\frac{n}{n+12}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse what should go in the numerator versus denominator of the probability fraction.

Some students think probability means comparing nonfiction books to fiction books only, leading them to write \(\frac{n}{12}\). This misses the fundamental concept that probability compares favorable outcomes to ALL possible outcomes.

This may lead them to select Choice A (\(\frac{n}{12}\))

Second Most Common Error:

Weak TRANSLATE skill: Students flip the probability fraction, putting total outcomes in the numerator instead of favorable outcomes.

They might think "there are more total books than nonfiction books, so the bigger number should be on top," leading to \(\frac{n+12}{n}\). When this doesn't appear as a choice, they might select the closest-looking fraction.

This causes confusion and may lead them to select Choice C (\(\frac{12}{n}\)) or guess randomly.

The Bottom Line:

This problem tests whether students truly understand what probability means - it's always favorable outcomes divided by total possible outcomes, not favorable compared to unfavorable or any other comparison.