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, the key principle is that the probability of both events occurring together (the intersection of the events) must equal the product of their individual probabilities. This relationship is written mathematically as: P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)P(A∩B)=P(A)×P(B)
In this case:
- P(A)=0.60P(A) = 0.60P(A)=0.60 is the probability that a shopper buys bread
- P(B)=0.20P(B) = 0.20P(B)=0.20 is the probability that a shopper buys cheese
If the two events are independent, then the probability that a shopper buys both bread and cheese is calculated as: P(A∩B)=0.60×0.20=0.12P(A \cap B) = 0.60 \times 0.20 = 0.12P(A∩B)=0.60×0.20=0.12
This is the condition for independence. Therefore, the events are independent if the probability of both buying bread and buying cheese is 0.12. Option C matches this result.
Now let us consider the other options and why they are incorrect:
- Option A says the probability of buying bread or cheese is 0.12. This value does not relate directly to independence. In fact, the probability of A or B is generally calculated by: P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) – P(A \cap B)P(A∪B)=P(A)+P(B)−P(A∩B) Using this formula, if the events are independent: P(A∪B)=0.60+0.20−0.12=0.68P(A \cup B) = 0.60 + 0.20 – 0.12 = 0.68P(A∪B)=0.60+0.20−0.12=0.68 So 0.12 is incorrect for the union.
- Option B says the probability of buying both is 0, which would mean the events are mutually exclusive, not independent.
- Option D suggests that the probability of buying bread or cheese is 0.80, which again does not indicate independence by itself. Based on earlier calculation, 0.68 would be the correct value if the events are independent.
So, only Option C correctly represents the condition for independence between buying bread and buying cheese.
