Each vertex of a 14-sided polygon is labeled with one of the 14 letters A through N, with a different...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
Each vertex of a 14-sided polygon is labeled with one of the 14 letters A through N, with a different letter at each vertex. If one vertex is selected at random, what is the probability that the letter D will be at the selected vertex? (Express your answer as a decimal or fraction, not as a percent.)
1. TRANSLATE the problem information
- Given information:
- 14-sided polygon (so 14 vertices total)
- Each vertex labeled with letters A through N (that's exactly 14 letters)
- Each vertex has a different letter
- One vertex selected at random
- What this tells us: Exactly one vertex has letter D, and there are 14 total vertices to choose from.
2. INFER the approach
- This is a basic probability problem asking for P(selected vertex has letter D)
- We need to use the fundamental probability formula: \(\mathrm{P(event)} = \frac{\mathrm{favorable\ outcomes}}{\mathrm{total\ outcomes}}\)
- Favorable outcomes = number of vertices with letter D = 1
- Total outcomes = total number of vertices = 14
3. Calculate the probability
\(\mathrm{P(vertex\ has\ letter\ D)} = \frac{1}{14}\)
To convert to decimal: \(1 \div 14 = 0.0714...\)
Answer: 1/14 or 0.0714
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misunderstand "each vertex is labeled with one of the 14 letters A through N, with a different letter at each vertex."
They might think this means multiple vertices could have the same letter, or they might think some letters don't appear at all. This leads to incorrect counting of favorable outcomes or total outcomes.
This confusion causes them to abandon systematic solution and guess randomly.
Second Most Common Error:
Missing conceptual knowledge: Students don't remember or properly apply the basic probability formula.
They might try to overcomplicate the problem, thinking about arrangements or permutations instead of recognizing this as simple probability. Some might calculate \(\frac{14}{1}\) instead of \(\frac{1}{14}\), getting a probability greater than 1.
This leads to confusion and guessing.
The Bottom Line:
This problem tests whether students can correctly interpret a straightforward probability setup and apply the most basic probability formula. The key insight is recognizing that when each vertex has a different letter, exactly one vertex has any particular letter.