A taste test asks people from Texas and California which pasta they prefer, brand $A$ or brand $B$. This table shows the results.
| | Brand A | Brand B | Total |
|---|---|---|---|
| Texas | 80 | 45 | 125 |
| California | 96 | 54 | 150 |
| Total | 176 | 99 | 275 |
A person is randomly selected from those tested.
Are being from California and preferring brand B independent events? Why or why not?
A. No, they are not independent because $P(\text{California})=0.55$ and $P(\text{California}|\text{brand }B)=0.36$.
B. Yes, they are independent because $P(\text{California}) \approx 0.55$ and $P(\text{California}|\text{brand }B)=0.55$.
C. No, they are not independent because $P(\text{California})=0.55$ and $P(\text{California} \mid \text{brand }B)=0.55$.
D. Yes, they are independent because $P(\text{California})=0.55$ and $P(\text{California}|\text{brand }B) \approx 0.36$.
Answer (1)
Calculate P ( C a l i f or nia ) = 275 150 ≈ 0.5455 .
Calculate P ( C a l i f or nia ∣ B r an d B ) = 99 54 ≈ 0.5455 .
Compare the probabilities: P ( C a l i f or nia ) = P ( C a l i f or nia ∣ B r an d B ) .
Since the probabilities are equal, the events are independent. Y es
Explanation
Analyze the problem and data We are given a table showing pasta preferences of people from Texas and California. We need to determine if the events 'being from California' and 'preferring brand B' are independent. Two events are independent if the probability of one event occurring does not affect the probability of the other event occurring. Mathematically, events A and B are independent if P ( A ∣ B ) = P ( A ) .
Calculate P(California) First, we calculate the probability of a person being from California. P ( C a l i f or nia ) = Total number of people Number of people from California = 275 150 = 11 6 P ( C a l i f or nia ) ≈ 0.5455
Calculate P(California | Brand B) Next, we calculate the conditional probability of a person being from California given that they prefer brand B. P ( C a l i f or nia ∣ B r an d B ) = Number of people who prefer brand B Number of people from California who prefer brand B = 99 54 = 11 6 P ( C a l i f or nia ∣ B r an d B ) ≈ 0.5455
Compare the probabilities and conclude Now we compare P ( C a l i f or nia ) and P ( C a l i f or nia ∣ B r an d B ) .
Since P ( C a l i f or nia ) = 11 6 ≈ 0.5455 and P ( C a l i f or nia ∣ B r an d B ) = 11 6 ≈ 0.5455 , we can see that P ( C a l i f or nia ) = P ( C a l i f or nia ∣ B r an d B ) .
Therefore, the events 'being from California' and 'preferring brand B' are independent.
Final Answer Based on our calculations, P ( C a l i f or nia ) ≈ 0.55 and P ( C a l i f or nia ∣ B r an d B ) ≈ 0.55 . Therefore, the events are independent. The correct answer is B.
Examples
In marketing, determining if customer demographics (like location) are independent of product preference helps target advertising effectively. If preference for a product is independent of location, a national campaign might be suitable. If not, marketing strategies should be tailored to specific regions.
