A school district conducted standardized testing across two grade levels. Grade 10 has 4 classes, with each class having 10...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A school district conducted standardized testing across two grade levels. Grade 10 has 4 classes, with each class having 10 students who passed the test and 5 students who failed the test. Grade 11 has 3 classes, with each class having 12 students who passed the test and 8 students who failed the test. A student will be selected at random from all students who took the test. What is the probability of selecting a Grade 11 student, given that the selected student passed the test?
1. TRANSLATE the problem information
- Given information:
- Grade 10: 4 classes, each with 10 passed and 5 failed
- Grade 11: 3 classes, each with 12 passed and 8 failed
- Need: \(\mathrm{P(Grade\,11\,student\,|\,student\,passed\,the\,test)}\)
- The phrase "given that the selected student passed" signals conditional probability
2. VISUALIZE the data in organized categories
- Calculate totals for each group:
- Grade 10 passed: \(\mathrm{4 \times 10 = 40}\) students
- Grade 10 failed: \(\mathrm{4 \times 5 = 20}\) students
- Grade 11 passed: \(\mathrm{3 \times 12 = 36}\) students
- Grade 11 failed: \(\mathrm{3 \times 8 = 24}\) students
3. INFER the solution approach
- This is conditional probability: \(\mathrm{P(Grade\,11\,|\,passed)}\)
- We need: (Grade 11 students who passed) ÷ (All students who passed)
- The "failed" students don't matter since we're only considering those who passed
4. SIMPLIFY the calculation
- Total students who passed: \(\mathrm{40 + 36 = 76}\) students
- Grade 11 students who passed: 36 students
- P(Grade 11 | passed) = \(\mathrm{36/76 = 9/19}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misunderstand what "given that the selected student passed" means and calculate \(\mathrm{P(Grade\,11)}\) instead of \(\mathrm{P(Grade\,11\,|\,passed)}\).
They calculate: (Total Grade 11 students) ÷ (All students) = \(\mathrm{(36 + 24) \div (40 + 20 + 36 + 24) = 60/120 = 1/2}\)
Since \(\mathrm{1/2}\) isn't among the choices, this leads to confusion and guessing.
Second Most Common Error:
Poor INFER reasoning: Students recognize it's conditional probability but set up the wrong fraction, calculating (All Grade 11) ÷ (Grade 11 who passed) = \(\mathrm{60/36 = 5/3}\), which is impossible for probability.
This causes them to get stuck and randomly select an answer.
The Bottom Line:
The key challenge is recognizing that "given that the student passed" restricts our sample space to only the 76 students who passed, not all 120 students who took the test.