A marketing firm tracks data on grocery store visits. In one study, it finds that the probability that a shopper buys bread during a visit to the grocery store is 0.60, and the probability that a shopper buys cheese is 0.20. Event A = A shopper buys bread. Event B = A shopper buys cheese. A and B are independent events if _____ A. the probability of buying bread or cheese is 0.12 B. the probability of buying bread and cheese is 0 C. the probability of buying bread and cheese is 0.12 D. the probability of buying bread or cheese is 0.80 SUBMIT
The Correct Answer and Explanation is:
Correct Answer: C. the probability of buying bread and cheese is 0.12
To determine whether two events are independent, consider the definition of independence in probability. Two events, A and B, are said to be independent if the occurrence of one does not affect the probability of the other occurring. Mathematically, this means that:
P(A and B) = P(A) × P(B)
In this problem, the marketing firm provides the following data:
- Probability that a shopper buys bread, P(A) = 0.60
- Probability that a shopper buys cheese, P(B) = 0.20
If these two events are independent, the probability that a shopper buys both bread and cheese would be:
P(A and B) = 0.60 × 0.20 = 0.12
Thus, the condition for A and B to be independent is that the probability of both occurring together equals 0.12. This matches option C, making it the correct choice.
Let’s briefly consider why the other options are incorrect:
- Option A: the probability of buying bread or cheese is 0.12. This is incorrect. The formula for the probability of A or B is P(A or B) = P(A) + P(B) – P(A and B). Using the given data and assuming independence, this becomes 0.60 + 0.20 – 0.12 = 0.68, not 0.12.
- Option B: the probability of buying bread and cheese is 0. This would mean the events are mutually exclusive, not independent.
- Option D: the probability of buying bread or cheese is 0.80. That would imply P(A and B) = 0.60 + 0.20 – 0.80 = 0.00, which contradicts the expected value of 0.12 under independence.
Therefore, the correct and logically supported answer is C.
