Understanding Candidate Failure Rates in Exams

This problem involves analyzing the results of an examination where 500 candidates took tests in three subjects: English, Hindi, and Mathematics. We are given the number of candidates who failed in specific combinations of these subjects, and we need to find the percentage of candidates who failed in at least two subjects.

Given Data on Candidate Failures

Let's list the failure data provided:

• Total candidates: 500
• Failed in English only (E only): 30
• Failed in Hindi only (H only): 75
• Failed in Mathematics only (M only): 50
• Failed in English and Hindi (E & H): 15
• Failed in Hindi and Mathematics (H & M): 17
• Failed in Mathematics and English (M & E): 17
• Failed in English, Hindi, and Mathematics (E & H & M): 5

Defining "Failed in at Least Two Subjects"

Failing in "at least two subjects" means a candidate failed in either exactly two subjects or in all three subjects. To find the total number of candidates who failed in at least two subjects, we need to sum up the candidates who failed in:

• Exactly English and Hindi
• Exactly Hindi and Mathematics
• Exactly Mathematics and English
• Exactly English, Hindi, and Mathematics (which is failing in all three)

Calculating Candidates Failing in Exactly Two Subjects

The given numbers for "Failed in English and Hindi" (15), "Failed in Hindi and Mathematics" (17), and "Failed in Mathematics and English" (17) include those who failed in all three subjects (5). To find the number who failed in *exactly* two subjects, we must subtract the number who failed in all three from these intersection values.

• Failed in Exactly English and Hindi = (Failed in E & H) - (Failed in E & H & M)
• Failed in Exactly English and Hindi = $15 - 5 = 10$
• Failed in Exactly Hindi and Mathematics = (Failed in H & M) - (Failed in E & H & M)
• Failed in Exactly Hindi and Mathematics = $17 - 5 = 12$
• Failed in Exactly Mathematics and English = (Failed in M & E) - (Failed in E & H & M)
• Failed in Exactly Mathematics and English = $17 - 5 = 12$

Total Candidates Failing in At Least Two Subjects

Now, we sum the numbers of candidates who failed in exactly two subjects and those who failed in all three subjects:

Total failed in at least two subjects = (Failed in Exactly E & H) + (Failed in Exactly H & M) + (Failed in Exactly M & E) + (Failed in E & H & M)

Total failed in at least two subjects = $10 + 12 + 12 + 5 = 39$

So, 39 candidates failed in at least two subjects.

Calculating the Percentage of Candidates Failing in At Least Two Subjects

To find the percentage, we divide the number of candidates who failed in at least two subjects by the total number of candidates and multiply by 100.

Percentage = $\left( \frac{\text{Number of candidates failed in at least two subjects}}{\text{Total number of candidates}} \right) \times 100$

Percentage = $\left( \frac{39}{500} \right) \times 100$

Percentage = $0.078 \times 100$

Percentage = $7.8\%$

Therefore, the percentage of candidates who failed in at least two subjects is 7.8%.

Revision Table: Exam Failure Analysis

Here is a quick summary of the calculations for understanding exam failure rates:

Additional Information on Set Theory in Data Analysis

Problems like this one, involving overlaps between different categories (in this case, failures in different subjects), can be effectively solved using principles from set theory, specifically the inclusion-exclusion principle. While we didn't explicitly use the full inclusion-exclusion formula for the union of three sets here, the calculation for "exactly two subjects" is derived directly from understanding set intersections and how the triple intersection (all three subjects) is counted within the pairwise intersections.

• Set Theory: A branch of mathematics dealing with collections of objects (sets).
• Intersection: The set of elements common to two or more sets (e.g., candidates failing in both English and Hindi).
• Union: The set of all elements in any of the given sets (e.g., candidates failing in English OR Hindi OR Mathematics).
• Inclusion-Exclusion Principle: A technique used to count the number of elements in a union of multiple sets by summing the sizes of individual sets, subtracting the sizes of all pairwise intersections, adding back the size of the triple intersection, and so on.

In this specific problem, to find candidates failing in *at least two* subjects, we directly calculated the sum of those failing in *exactly two* pairs plus those failing in *all three*. This approach is simpler for this specific question than calculating the full union or other complex combinations.