GMAT Focus EditionQuantitative ReasoningMedium
A market research firm conducted a survey on product preference. 70% of respondents preferred Product A, 50% preferred Product B, and 30% preferred both Product A and Product B. What percentage of respondents preferred neither Product A nor Product B?
- A10%
- B20%
- C30%
- D40%
Show answer & explanationAnswer & explanation
Correct answer: A. 10%
This is a set theory problem (Venn diagrams). Use the formula: P(A or B) = P(A) + P(B) - P(A and B). P(A or B) = 70% + 50% - 30% = 90%. The percentage that preferred neither is 100% - P(A or B) = 100% - 90% = 10%.
Why the other options are wrong
- B. This might result from incorrectly calculating P(A or B) or a simple subtraction error.
- C. This is the percentage that preferred both, not neither.
- D. This might result from simply subtracting the 'both' from 100% without accounting for the overlaps.
Inclusion-Exclusion Principle (Two Sets)
For two sets A and B, the number of elements in their union is the sum of the elements in each set minus the number of elements in their intersection.
- Used to find the size of the union of two sets.
- Formula: |A U B| = |A| + |B| - |A ∩ B|.
- Can be extended to probabilities: P(A U B) = P(A) + P(B) - P(A ∩ B).
Memory trick: When counting, don't double-count the overlap; subtract it once.