To determine whether being from California and preferring brand B are independent events, we need to compare two probabilities:
1. The probability of being from California, [tex]\( P(\text{California}) \)[/tex].
2. The probability of being from California given that a person prefers brand B, [tex]\( P(\text{California} \mid \text{brand B}) \)[/tex].
Let's break down the calculations step by step.
### Step 1: Calculate [tex]\( P(\text{California}) \)[/tex]
From the table, the total number of people tested is 275, and the number of people from California is 150. Therefore,
[tex]\[ P(\text{California}) = \frac{\text{Number of people from California}}{\text{Total number of people tested}} = \frac{150}{275} \approx 0.545 \][/tex]
### Step 2: Calculate [tex]\( P(\text{California} \mid \text{brand B}) \)[/tex]
To find [tex]\( P(\text{California} \mid \text{brand B}) \)[/tex], we need to find the number of people who prefer brand B and are from California, and divide it by the total number of people who prefer brand B.
From the table:
- Total number of people who prefer brand B: 99
- Number of people who prefer brand B and are from California: 54
Therefore,
[tex]\[ P(\text{California} \mid \text{brand B}) = \frac{\text{Number of people from California and prefer brand B}}{\text{Total number of people who prefer brand B}} = \frac{54}{99} \approx 0.545 \][/tex]
### Step 3: Compare the probabilities
- [tex]\( P(\text{California}) \approx 0.545 \)[/tex]
- [tex]\( P(\text{California} \mid \text{brand B}) \approx 0.545 \)[/tex]
### Step 4: Conclusion
Since [tex]\( P(\text{California}) \approx P(\text{California} \mid \text{brand B}) \)[/tex], the events "being from California" and "preferring brand B" are independent.
Therefore, the correct answer is:
D. 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].
