A box contains 9 pencils, 7 of which are red and the rest are blue. If a pencil is selected...

1. TRANSLATE the problem information

• Given information:
  • Box contains 9 pencils total
  • 7 pencils are red
  • The rest are blue
  • Need to find probability of selecting blue

2. INFER what 'the rest are blue' means

• If 7 out of 9 pencils are red, then the remaining pencils must be blue
• Number of blue pencils = \(9 - 7 = 2\)

3. INFER the probability approach

• Use basic probability formula: \(\mathrm{P(event)} = \frac{\mathrm{favorable\ outcomes}}{\mathrm{total\ outcomes}}\)
• For blue pencil: favorable outcomes = 2 blue pencils, total outcomes = 9 total pencils

4. Calculate the probability

• \(\mathrm{P(blue)} = \frac{2}{9}\)

Answer: B (2/9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students mix up which quantity represents favorable outcomes vs. total outcomes.

They see '7 red pencils' prominently mentioned and mistakenly use this as the numerator, thinking \(\mathrm{P(blue)} = \frac{7}{9}\). They fail to recognize that the question asks for blue pencils, not red ones.

This leads them to select Choice C (7/9).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand 'the rest are blue' and don't perform the subtraction step.

They might think there's only 1 blue pencil (perhaps confusing this with a different setup), leading to \(\mathrm{P(blue)} = \frac{1}{9}\).

This may lead them to select Choice A (1/9).

The Bottom Line:

This problem tests whether students can correctly identify the favorable outcomes when the desired event is the complement of what's explicitly stated. The key insight is translating 'the rest are blue' into a concrete number through subtraction.