Digital SATMath: Problem-Solving and Data AnalysisMedium
A market researcher observed that 45% of customers in a store purchased item A, and 30% purchased item B. If 15% of customers purchased both item A and item B, what percentage of customers purchased neither item A nor item B?
- A75%
- B40%
- C10%
- D20%
Show answer & explanationAnswer & explanation
Correct answer: B. 40%
Using the principle of inclusion-exclusion: P(A or B) = P(A) + P(B) - P(A and B). P(A or B) = 45% + 30% - 15% = 60%. The percentage that purchased neither is 100% - P(A or B) = 100% - 60% = 40%.
Why the other options are wrong
- A. This is the sum of P(A) and P(B) without accounting for the overlap, or perhaps the total of those who bought at least one item (45+30 = 75) if overlap wasn't subtracted.
- C. This might result from incorrectly calculating the percentage that bought at least one item (e.g., 45+30-15 = 60, then 100-60 = 40).
- D. This might be 100% - (45% + 30% + 15%) = 100 - 90 = 10, or some other miscalculation.
Probability (Inclusion-Exclusion)
The Principle of Inclusion-Exclusion is used to find the probability of the union of two events by summing individual probabilities and subtracting the probability of their intersection to avoid double-counting.
- P(A or B) = P(A) + P(B) - P(A and B).
- Used when events are not mutually exclusive.
- The sum of P(A or B) and P(neither A nor B) is 1 (or 100%).
Memory trick: Add the circles, then take out the middle overlap, then subtract from total.