To determine whether [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent events, we need to compare [tex]\( P(Z) \)[/tex] and [tex]\( P(Z \mid B) \)[/tex]. Let's proceed step by step:
1. Calculate [tex]\( P(Z) \)[/tex]:
[tex]\[
P(Z) = \frac{\text{Total number of \( Z \) outcomes}}{\text{Total number of outcomes}}
\][/tex]
Given:
[tex]\[
\text{Total number of } Z \text{ outcomes} = 297
\][/tex]
[tex]\[
\text{Total number of outcomes} = 660
\][/tex]
Thus:
[tex]\[
P(Z) = \frac{297}{660} \approx 0.45
\][/tex]
2. Calculate [tex]\( P(B) \)[/tex]:
[tex]\[
P(B) = \frac{\text{Total number of \( B \) outcomes}}{\text{Total number of outcomes}}
\][/tex]
Given:
[tex]\[
\text{Total number of } B \text{ outcomes} = 280
\][/tex]
[tex]\[
P(B) = \frac{280}{660} \approx 0.42424242424242425
\][/tex]
3. Calculate [tex]\( P(Z \mid B) \)[/tex]:
[tex]\[
P(Z \mid B) = \frac{\text{Number of outcomes that are both \( Z \) and \( B \)}}{\text{Total number of \( B \) outcomes}}
\][/tex]
Given:
[tex]\[
\text{Number of outcomes that are both } Z \text{ and } B = 126
\][/tex]
[tex]\[
\text{Total number of } B \text{ outcomes} = 280
\][/tex]
Thus:
[tex]\[
P(Z \mid B) = \frac{126}{280} \approx 0.45
\][/tex]
4. Compare [tex]\( P(Z) \)[/tex] and [tex]\( P(Z \mid B) \)[/tex]:
[tex]\[
P(Z) = 0.45
\][/tex]
[tex]\[
P(Z \mid B) = 0.45
\][/tex]
Since:
[tex]\[
P(Z) = P(Z \mid B)
\][/tex]
We can conclude that [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent events because the probability of [tex]\( Z \)[/tex] occurring is the same whether or not [tex]\( B \)[/tex] occurs.
Therefore, the true statement is:
[tex]\[
\text{Z and B are independent events because } P(Z \mid B) = P(Z).
\][/tex]
