A taste test asks people from Texas and California which pasta they prefer, brand A or brand B. The table shows the results.

[tex]\[
\begin{tabular}{|l|c|c|c|}
\hline
& Brand A & Brand B & Total \\
\hline
Texas & 80 & 45 & 125 \\
\hline
California & 96 & 54 & 150 \\
\hline
Total & 176 & 99 & 275 \\
\hline
\end{tabular}
\][/tex]

A person is randomly selected from those tested.

Are being from California and preferring brand B independent events? Why or why not?

A. Yes, they are independent because [tex]\(P(\text{California}) \approx 0.55\)[/tex] and [tex]\(P(\text{California} \mid \text{brand B}) \approx 0.55\)[/tex].

B. No, they are not independent because [tex]\(P(\text{California}) \approx 0.55\)[/tex] and [tex]\(P(\text{California} \mid \text{brand B}) \approx 0.36\)[/tex].

C. Yes, they are independent because [tex]\(P(\text{California}) \approx 0.55\)[/tex] and [tex]\(P(\text{California} \mid \text{brand B}) \approx 0.36\)[/tex].

D. No, they are not independent because [tex]\(P(\text{California}) \approx 0.55\)[/tex] and [tex]\(P(\text{California} \mid \text{brand B}) \approx 0.55\)[/tex].



Answer :

To determine whether being from California and preferring brand B are independent events, we need to check if the probability of being from California and preferring brand B equals the product of the individual probabilities of being from California and preferring brand B. Let us break it down step by step:

1. Calculate [tex]\( P(\text{California}) \)[/tex]:
- The total number of people surveyed is 275.
- The number of people from California is 150.
- Therefore, [tex]\( P(\text{California}) = \frac{150}{275} \approx 0.545 \)[/tex].

2. Calculate [tex]\( P(\text{Brand B}) \)[/tex]:
- The total number of people who prefer Brand B is 99.
- Therefore, [tex]\( P(\text{Brand B}) = \frac{99}{275} \approx 0.36 \)[/tex].

3. Calculate [tex]\( P(\text{California and Brand B}) \)[/tex]:
- The number of people from California who prefer Brand B is 54.
- Therefore, [tex]\( P(\text{California and Brand B}) = \frac{54}{275} \approx 0.196 \)[/tex].

4. Calculate [tex]\( P(\text{California} | \text{Brand B}) \)[/tex]:
- The number of people who prefer Brand B is 99.
- The number of those who are from California and prefer Brand B is 54.
- Therefore, [tex]\( P(\text{California} | \text{Brand B}) = \frac{54}{99} \approx 0.545 \)[/tex].

5. Check for Independence:
- Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if [tex]\( P(A \text{ and } B) = P(A) \cdot P(B) \)[/tex].
- Calculate [tex]\( P(\text{California}) \times P(\text{Brand B}) \)[/tex]:
[tex]\[ P(\text{California}) \times P(\text{Brand B}) = 0.545 \times 0.36 \approx 0.196. \][/tex]
- Compare this with [tex]\( P(\text{California and Brand B}) \)[/tex]:
[tex]\[ P(\text{California and Brand B}) \approx 0.196. \][/tex]
- Since [tex]\( P(\text{California}) \times P(\text{Brand B}) \approx P(\text{California and Brand B}) \)[/tex], the events are independent.

Therefore, the correct option is:
A. Yes, they are independent because [tex]\( P(\text{California}) \approx 0.55 \)[/tex] and [tex]\( P(\text{California} | \text{Brand B}) \approx 0.55 \)[/tex].