In a group of 11 people, 4 have brown eyes. If a person from this group is selected at random,...

1. TRANSLATE the problem information

• Given information:
  • Total people in the group: 11
  • People with brown eyes: 4
  • Need: Probability of selecting someone with brown eyes

• What this tells us: We have all the numbers needed for a basic probability calculation

2. INFER the approach

• This is asking for basic probability: the chance of getting our desired outcome
• We need the probability formula: \(\mathrm{P(event)} = \frac{\mathrm{favorable\ outcomes}}{\mathrm{total\ outcomes}}\)
• Favorable outcomes = people with brown eyes = 4
• Total outcomes = all people who could be selected = 11

3. Apply the probability formula

• \(\mathrm{P(brown\ eyes)} = \frac{4}{11}\)

Answer: B) 4/11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students correctly identify the total (11) and the brown-eyed people (4), but then calculate the probability of selecting someone WITHOUT brown eyes instead.

They think: "7 people don't have brown eyes out of 11 total, so \(\mathrm{P} = \frac{7}{11}\)"

This may lead them to select Choice D (7/11)

Second Most Common Error:

Conceptual confusion about probability setup: Students get confused about what the denominator should be and use only the non-brown eyed people as the total.

They think: "4 people with brown eyes compared to 7 people without brown eyes, so \(\mathrm{P} = \frac{4}{7}\)"

This may lead them to select Choice C (4/7)

The Bottom Line:

The key insight is clearly identifying what counts as your "favorable outcome" versus your "total possible outcomes." In probability, the denominator is always the total number of items you're choosing from, not just the "other" category.