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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 Texas and preferring brand A independent events? Why or why not?

A. No, they are not independent because [tex]$P(\text{Texas}) \approx 0.45$[/tex] and [tex]$P(\text{Texas} \mid \text{brand A}) \approx 0.45$[/tex].
B. No, they are not independent because [tex]$P(\text{Texas}) = 0.45$[/tex] and [tex]$P(\text{Texas} \mid \text{brand A}) \approx 0.64$[/tex].

Sagot :

GinnyAnswer
To determine if the events "being from Texas" and "preferring brand A" are independent, we need to compare the probability of being from Texas [tex]\( P(\text{Texas}) \)[/tex] with the conditional probability of being from Texas given preferring brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]. If these two probabilities are equal, the events are independent.

Here are the steps to find these probabilities and determine if the events are independent:

### Step 1: Calculate the Probability of Being from Texas [tex]\( P(\text{Texas}) \)[/tex]
The total number of participants is 275.

The number of participants from Texas is 125.

[tex]\[ P(\text{Texas}) = \frac{\text{Number of participants from Texas}}{\text{Total number of participants}} = \frac{125}{275} \approx 0.4545 \][/tex]

### Step 2: Calculate the Probability of Preferring Brand A [tex]\( P(\text{Brand A}) \)[/tex]
The number of participants who prefer Brand A is 176.

[tex]\[ P(\text{Brand A}) = \frac{\text{Number of participants who prefer Brand A}}{\text{Total number of participants}} = \frac{176}{275} \approx 0.64 \][/tex]

### Step 3: Calculate the Conditional Probability of Being from Texas Given Preferring Brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]
The number of participants from Texas who prefer Brand A is 80.

The total number of participants who prefer Brand A is 176.

[tex]\[ P(\text{Texas} \mid \text{Brand A}) = \frac{\text{Number of participants from Texas who prefer Brand A}}{\text{Total number of participants who prefer Brand A}} = \frac{80}{176} \approx 0.4545 \][/tex]

### Step 4: Compare Probabilities to Determine Independence
Compare [tex]\( P(\text{Texas}) \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]:

[tex]\[ P(\text{Texas}) = 0.4545 \][/tex]

[tex]\[ P(\text{Texas} \mid \text{Brand A}) = 0.4545 \][/tex]

Since [tex]\( P(\text{Texas}) \approx P(\text{Texas} \mid \text{Brand A}) \)[/tex], the events "being from Texas" and "preferring brand A" are independent.

### Conclusion
The correct answer is:
A. No, they are not independent because [tex]\( P(\text{Texas}) \approx 0.45 \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.45 \)[/tex].
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