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

\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}

A person is randomly selected from those tested. Are being from California and preferring brand A independent events? Why or why not?

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

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

C. No, they are not independent because [tex]\( P(\text{California}) \approx 0.55 \)[/tex] and [tex]\( P(\text{California|brand A}) \approx 0.64 \)[/tex].

D. Yes, they are independent because [tex]\( P(\text{California}) \approx 0.55 \)[/tex] and [tex]\( P(\text{California|brand A}) = 0.64 \)[/tex].



Answer :

GinnyAnswer
To determine whether being from California and preferring brand [tex]$A$[/tex] are independent events, we need to compare two probabilities:

1. [tex]\( P(\text{California}) \)[/tex]: The probability that a randomly selected person is from California.
2. [tex]\( P(\text{California} \mid \text{brand A}) \)[/tex]: The probability that a person is from California given that they prefer brand [tex]$A$[/tex].

If these two probabilities are equal, the events are independent. Otherwise, they are not.

### Step-by-Step Solution:

1. Calculate [tex]\( P(\text{California}) \)[/tex]:
[tex]\[ P(\text{California}) = \frac{\text{Number of California people}}{\text{Total number of people}} = \frac{150}{275} \approx 0.5454545454545454 \][/tex]

2. Calculate [tex]\( P(\text{California} \mid \text{brand A}) \)[/tex]:
[tex]\[ P(\text{California} \mid \text{brand A}) = \frac{\text{Number of people from California who prefer brand A}}{\text{Total number of people who prefer brand A}} = \frac{96}{176} \approx 0.5454545454545454 \][/tex]

3. Compare the probabilities:
[tex]\[ P(\text{California}) \approx 0.55 \][/tex]
[tex]\[ P(\text{California} \mid \text{brand A}) \approx 0.55 \][/tex]

Since [tex]\( P(\text{California}) \)[/tex] is approximately equal to [tex]\( P(\text{California} \mid \text{brand A}) \)[/tex], the two events (being from California and preferring brand A) are independent.

### Conclusion:

The correct answer is:

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

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