Determining the Number of Students Who Passed Both Mathematics and English
In a recent study involving 15,000 candidates, a detailed examination of their performance in Mathematics and English revealed interesting data. This article aims to explore the problem of determining how many students passed in both Mathematics and English given specific fail rates in each subject. We will employ the principle of inclusion-exclusion and probability to solve the problem effectively.
The Problem and Data
The problem at hand is as follows: in an examination of 15,000 candidates, 35% (3,500) failed in Mathematics and 38% (3,800) failed in English, with 28% (2,800) passing in both subjects. The goal is to determine the number of students who failed in both subjects.
Applying the Inclusion-Exclusion Principle
Let's define the sets:
A Set of students who failed in Mathematics
B Set of students who failed in English
Using the principle of inclusion-exclusion, we can express the union of sets A and B (students who failed in either Mathematics or English or both) as:
N(A ∪ B) N(A) N(B) - N(A ∩ B)
Where:
3,500 students failed in Mathematics (N(A) 3,500) 3,800 students failed in English (N(B) 3,800) 2,800 students passed in both subjects (N(A ∩ B) 15,000 - 2,800 12,200)Substituting these values into the formula, we get:
15,000 3,500 3,800 - N(A ∩ B)
Solving for N(A ∩ B):
N(A ∩ B) 3,500 3,800 - 15,000 7,300 - 15,000 1,000
Thus, 1,000 students failed in both Mathematics and English.
Verifying the Solution
To verify, let's check the conditional probabilities:
P(A) 0.35 (3,500 out of 15,000) P(B) 0.38 (3,800 out of 15,000) P(A ∩ B) 0.28 (2,800 out of 15,000)The probability of failing at least one subject (A ∪ B) is calculated as:
P(A ∪ B) P(A) P(B) - P(A ∩ B)
Substituting the known values:
P(A ∪ B) 0.35 0.38 - 0.28 0.45 (4,500 out of 15,000)
Thus, the number of students who passed in both subjects is:
N(A ∩ B) 15,000 - 4,500 10,500
So, 10,500 out of 15,000 students passed in both Mathematics and English.
Conclusion
Using the principle of inclusion-exclusion and probability, we determined that 1,000 students failed in both Mathematics and English, while 10,500 passed in both subjects. This analysis helps in understanding the performance distribution among students and can be useful in educational and training evaluations.