A taste test asks people from Texas and California which pasta they prefer, brand A or brand B. This 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 A independent events? Why or why not?

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

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].

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

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



Answer :

To determine whether the events "being from California" and "preferring brand A" are independent, we need to compare two probabilities:
1. The probability of being from California, [tex]\( P(\text{California}) \)[/tex].
2. The conditional probability of being from California given that the person prefers brand A, [tex]\( P(\text{California} \mid \text{Brand A}) \)[/tex].

Step 1: Calculate [tex]\( P(\text{California}) \)[/tex]

The total number of people surveyed is 275, and the number of people from California is 150. Therefore, the probability of being from California is:
[tex]\[ P(\text{California}) = \frac{\text{Number of people from California}}{\text{Total number of people}} = \frac{150}{275} \][/tex]

Simplifying this fraction:
[tex]\[ P(\text{California}) = \frac{150}{275} \approx 0.545 \][/tex]

Step 2: Calculate [tex]\( P(\text{California} \mid \text{Brand A}) \)[/tex]

Next, we need to find the conditional probability that a person is from California given they prefer brand A. This is given by the ratio of the number of people from California who prefer brand A to the total number of people who prefer brand A. According to the table, 96 people from California prefer brand A out of a total of 176 people who prefer brand A.

[tex]\[ P(\text{California} \mid \text{Brand A}) = \frac{\text{Number of people from California who prefer Brand A}}{\text{Total number of people who prefer Brand A}} = \frac{96}{176} \][/tex]

Simplifying this fraction:
[tex]\[ P(\text{California} \mid \text{Brand A}) = \frac{96}{176} \approx 0.545 \][/tex]

Step 3: Determine Dependence or Independence

For two events to be independent, the probability of one event must be equal to the conditional probability given the other event. In other words, [tex]\( P(\text{California}) \)[/tex] should equal [tex]\( P(\text{California} \mid \text{Brand A}) \)[/tex].

From the calculations:
[tex]\[ P(\text{California}) \approx 0.545 \][/tex]
[tex]\[ P(\text{California} \mid \text{Brand A}) \approx 0.545 \][/tex]

Since these two probabilities are approximately equal, the two events being from California and preferring Brand A are independent.

Therefore, 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].