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In a survey of 260 students in a school, 100 were listed as taking apple juice, 150 as taking orange juice and 75 were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.
75
85
65
95
85
Subtract the intersection (both) from the sum of individual sets to find the union, then subtract the union from the total pool: 260 - (100 + 150 - 75) = 260 - 175 = 85.
Total students (n(U)) = 260, students taking apple juice (n(A)) = 100, students taking orange juice (n(O)) = 150, students taking both (n(A \cap O)) = 75.
n(AcupO)=n(A)+n(O)тИТn(AcapO), n(Neither)=n(U)тИТn(AcupO)
Subtract the intersection (both) from the sum of individual sets to find the union, then subtract the union from the total pool: 260 - (100 + 150 - 75) = 260 - 175 = 85.
Adding n(A) and n(O) and subtracting the total (260) directly without accounting for the overlap (75), which results in double-counting the students who take both.
Find number of students taking at least one juice
Use the Principle of Inclusion-Exclusion to find n(AcupO).
n(AcupO)=100+150тИТ75=175
Calculate number of students taking neither
Subtract the result of the union from the total number of surveyed students.
n(Neither)=260тИТ175=85
B is correct because the total number of students drinking at least one juice is 175, leaving 85 students out of 260 who drink neither.
This concept of Inclusion-Exclusion is widely applicable in Probability and Data Interpretation questions involving overlapping categories.
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