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In a survey of 400 workers, 150 prefer tea, 200 prefer coffee, and 50 prefer both. How many workers prefer neither tea nor coffee?
100
150
200
250
100
Subtract the intersection from the sum of individual preferences (150+200тИТ50=300) and subtract this result from the total (400тИТ300=100).
Total workers = 400, workers preferring tea n(T) = 150, workers preferring coffee n(C) = 200, workers preferring both n(T тИй C) = 50
n(TтИкC)=n(T)+n(C)тИТn(TтИйC)
Subtract the intersection from the sum of individual preferences (150+200тИТ50=300) and subtract this result from the total (400тИТ300=100).
Many students add 150 and 200 and subtract from 400 without accounting for the 50 who prefer both, ignoring the inclusion-exclusion overlap.
Calculate Union of Sets
Use the Principle of Inclusion-Exclusion to find the number of workers who prefer at least one beverage.
n(TтИкC)=150+200тИТ50=300
Calculate Complement
Subtract the number of workers who like at least one beverage from the total survey population to find those who like neither.
n(Neither)=400тИТ300=100
A is correct because the total number of workers who prefer at least one beverage is 300, leaving 100 workers who prefer neither.
This principle is widely used in probability theory for calculating P(AтИкB)=P(A)+P(B)тИТP(AтИйB), which is a frequent topic in competitive quantitative aptitude sections.