To determine whether [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent events, we need to compare the conditional probability [tex]\( P(Z \mid B) \)[/tex] with the probability [tex]\( P(Z) \)[/tex].
First, we calculate the probability [tex]\( P(Z) \)[/tex]:
[tex]\[ P(Z) = \frac{\text{Total count for } Z}{\text{Total count}} = \frac{297}{660} \][/tex]
Next, we calculate the probability [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = \frac{\text{Total count for } B}{\text{Total count}} = \frac{280}{660} \][/tex]
Then, we determine the conditional probability [tex]\( P(Z \mid B) \)[/tex]:
[tex]\[ P(Z \mid B) = \frac{\text{Count for both } B \text{ and } Z}{\text{Total count for } B} = \frac{126}{280} \][/tex]
To confirm whether [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent events, we check if [tex]\( P(Z \mid B) = P(Z) \)[/tex].
We start by verifying if the calculated values for [tex]\( P(Z) \)[/tex] and [tex]\( P(Z \mid B) \)[/tex] are equal. According to the given information, these values match:
[tex]\[ P(Z) = \frac{297}{660} \approx 0.45 \][/tex]
[tex]\[ P(Z \mid B) = \frac{126}{280} \approx 0.45 \][/tex]
Since [tex]\( P(Z \mid B) = P(Z) \)[/tex], we conclude that [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent events.
Therefore, the correct statement is:
- [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent events because [tex]\( P(Z \mid B) = P(Z) \)[/tex].
