To determine whether events [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent, we need to examine their probabilities and determine if they meet the criterion for independence. Specifically, we need to check if the conditional probability [tex]\( P(Z \mid B) \)[/tex] is equal to the marginal probability [tex]\( P(Z) \)[/tex].
First, let's define the relevant probabilities:
1. Total Number of Observations: [tex]\( 660 \)[/tex]
2. Number of Observations in [tex]\( Z \)[/tex]: [tex]\( 297 \)[/tex]
3. Number of Observations in [tex]\( B \)[/tex]: [tex]\( 280 \)[/tex]
4. Number of Observations in both [tex]\( Z \)[/tex] and [tex]\( B \)[/tex]: [tex]\( 126 \)[/tex]
Next, we calculate the probabilities:
1. Calculate [tex]\( P(Z) \)[/tex]:
[tex]\[ P(Z) = \frac{\text{Number of Observations in } Z}{\text{Total Number of Observations}} = \frac{297}{660} \][/tex]
2. Calculate [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = \frac{\text{Number of Observations in } B}{\text{Total Number of Observations}} = \frac{280}{660} \][/tex]
3. Calculate [tex]\( P(Z \mid B) \)[/tex]:
[tex]\[ P(Z \mid B) = \frac{\text{Number of Observations in both } Z \text{ and } B}{\text{Number of Observations in } B} = \frac{126}{280} \][/tex]
To determine if [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent, we need to check if:
[tex]\[ P(Z \mid B) = P(Z) \][/tex]
After calculating these probabilities, we found that:
[tex]\[ P(Z \mid B) = P(Z) \][/tex]
Since we have determined that [tex]\( P(Z \mid B) \)[/tex] is equal to [tex]\( P(Z) \)[/tex], this means [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are indeed independent events according to the definition. Therefore, the correct statement is:
[tex]\[ \text{Z and B are independent events because } P(Z \mid B) = P(Z). \][/tex]
