A grove has 6 rows of birch trees and 5 rows of maple trees. Each row of birch trees has...

1. TRANSLATE the problem information

• Given information:
  • 6 rows of birch trees: each has 8 trees \(\geq 20\) feet and 6 trees \(\lt 20\) feet
  • 5 rows of maple trees: each has 9 trees \(\geq 20\) feet and 7 trees \(\lt 20\) feet
  • Want probability of selecting maple tree, given tree is \(\geq 20\) feet
• This tells us we need conditional probability: \(\mathrm{P(maple \mid tree \geq 20 \text{ feet})}\)

2. INFER the approach

• Since we're given that the selected tree is \(\geq 20\) feet, we only count trees in this category
• Formula: \(\mathrm{P(maple \mid \geq 20 \text{ feet})} = \frac{\text{maple trees } \geq 20 \text{ feet}}{\text{total trees } \geq 20 \text{ feet}}\)
• We need to calculate both the numerator and denominator

3. Calculate maple trees that are \(\geq 20\) feet

• 5 rows of maple × 9 trees \(\geq 20\) feet per row = 45 maple trees \(\geq 20\) feet

4. Calculate birch trees that are \(\geq 20\) feet

• 6 rows of birch × 8 trees \(\geq 20\) feet per row = 48 birch trees \(\geq 20\) feet

5. Calculate total trees \(\geq 20\) feet

• 45 maple + 48 birch = 93 total trees \(\geq 20\) feet

6. SIMPLIFY to get final probability

• \(\mathrm{P(maple \mid \geq 20 \text{ feet})} = \frac{45}{93}\)
• Divide both numerator and denominator by 3: \(\frac{45}{93} = \frac{15}{31}\)

Answer: C. \(\frac{15}{31}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing this is conditional probability and using all trees instead of just those \(\geq 20\) feet

Students calculate: (total maple trees) / (total trees in grove)

• Total maple: \(5 \text{ rows} \times (9 + 7) = 5 \times 16 = 80\) trees
• Total trees: \(6 \times (8 + 6) + 5 \times (9 + 7) = 6 \times 14 + 5 \times 16 = 84 + 80 = 164\) trees
• Wrong answer: \(\frac{80}{164} = \frac{20}{41}\)

This doesn't match any answer choice exactly, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Getting the right setup \(\left(\frac{45}{93}\right)\) but failing to simplify the fraction

Students correctly identify they need \(\frac{45}{93}\) but don't recognize this simplifies to \(\frac{15}{31}\), so they look for \(\frac{45}{93}\) among the choices and get confused when it's not there. This leads to abandoning systematic solution and guessing.

The Bottom Line:

The key insight is recognizing that "given that the tree is \(\geq 20\) feet" creates a restricted sample space—we're not selecting from all trees, only from those meeting this height condition.