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, use the definition of independence:
Two events A and B are independent if and only if
P(A and B) = P(A) × P(B)
In this problem:
- P(A) = probability that a shopper buys bread = 0.60
- P(B) = probability that a shopper buys cheese = 0.20
If events A and B are independent, then:
P(A and B) = 0.60 × 0.20 = 0.12
So, if the probability that a shopper buys both bread and cheese is 0.12, then the events are independent. This is exactly what option C states.
Now, look at the other options:
- Option A: “The probability of buying bread or cheese is 0.12” is incorrect. The probability of A or B is calculated by
P(A or B) = P(A) + P(B) − P(A and B)
If A and B are independent, then P(A or B) = 0.60 + 0.20 − 0.12 = 0.68. So, 0.12 is not correct. - Option B: “The probability of buying bread and cheese is 0” would imply that it is impossible to buy both at the same time. This would make A and B mutually exclusive, not independent.
- Option D: “The probability of buying bread or cheese is 0.80” is also incorrect. As shown, P(A or B) equals 0.68, not 0.80, under the independence assumption.
Therefore, the only correct condition for independence is when the joint probability equals the product of the individual probabilities, which makes option C the best choice.
