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 0.25. 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 0.15 B. the probability of buying bread and cheese is 0.15 C. the probability of buying bread and cheese is 0 D. the probability of buying bread or cheese is 0.85
The Correct Answer and Explanation is:
Correct Answer: B. the probability of buying bread and cheese is 0.15
To understand independence between two events, recall that two events are independent if the occurrence of one does not affect the probability of the other. Mathematically, events A and B are independent if:P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)P(A and B)=P(A)×P(B)
In this question:
- P(A)=0.60P(A) = 0.60P(A)=0.60 (the probability that a shopper buys bread)
- P(B)=0.25P(B) = 0.25P(B)=0.25 (the probability that a shopper buys cheese)
Using the multiplication rule for independent events:P(A and B)=0.60×0.25=0.15P(A \text{ and } B) = 0.60 \times 0.25 = 0.15P(A and B)=0.60×0.25=0.15
This means that if the events are independent, the probability of a shopper buying both bread and cheese must be 0.15. This matches option B, making it the correct answer.
Now examine why the other options are incorrect:
- Option A states that the probability of buying bread or cheese is 0.15. This is incorrect because it refers to the union of events, not their independence.
- Option C suggests the probability of buying both is 0. This would imply a complete lack of overlap, which contradicts the idea of independence.
- Option D provides a value for the probability of buying bread or cheese as 0.85. This figure is close to the correct union probability, but calculating the true value requires:
P(A or B)=P(A)+P(B)−P(A and B)=0.60+0.25−0.15=0.70P(A \text{ or } B) = P(A) + P(B) – P(A \text{ and } B) = 0.60 + 0.25 – 0.15 = 0.70P(A or B)=P(A)+P(B)−P(A and B)=0.60+0.25−0.15=0.70
So 0.85 is incorrect, and independence is not determined using the union.
In conclusion, only option B correctly uses the definition of independence through the multiplication rule.
