To determine if events [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent, we need to check if [tex]\( P(Z \mid B) = P(Z) \)[/tex]. In other words, we must see if the probability of [tex]\( Z \)[/tex] given [tex]\( B \)[/tex] is equal to the overall probability of [tex]\( Z \)[/tex].
Let's go through the steps to determine this:
1. Calculate the joint probability [tex]\( P(Z \cap B) \)[/tex]:
- From the table, the number of outcomes where both [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] occur is 126.
- The total number of outcomes is 660.
- Therefore, [tex]\( P(Z \cap B) = \frac{126}{660} \)[/tex].
2. Calculate the probability [tex]\( P(B) \)[/tex]:
- From the table, the number of outcomes where [tex]\( B \)[/tex] occurs is 280.
- The total number of outcomes is 660.
- Therefore, [tex]\( P(B) = \frac{280}{660} \)[/tex].
3. Calculate the conditional probability [tex]\( P(Z \mid B) \)[/tex]:
- [tex]\( P(Z \mid B) = \frac{P(Z \cap B)}{P(B)} \)[/tex].
- So, [tex]\( P(Z \mid B) = \frac{\frac{126}{660}}{\frac{280}{660}} = \frac{126}{280} \)[/tex].
4. Calculate the overall probability [tex]\( P(Z) \)[/tex]:
- From the table, the number of outcomes where [tex]\( Z \)[/tex] occurs is 297.
- The total number of outcomes is 660.
- Therefore, [tex]\( P(Z) = \frac{297}{660} \)[/tex].
5. Compare [tex]\( P(Z \mid B) \)[/tex] and [tex]\( P(Z) \)[/tex]:
- From our calculations:
[tex]\[
P(Z \mid B) = \frac{126}{280} = 0.45
\][/tex]
[tex]\[
P(Z) = \frac{297}{660} = 0.45
\][/tex]
As both [tex]\( P(Z \mid B) \)[/tex] and [tex]\( P(Z) \)[/tex] are equal to 0.45, these two values are indeed the same.
Therefore, [tex]\( Z \)[/tex] and [tex]\( B \)[/tex] are independent events because [tex]\( P(Z \mid B) = P(Z) \)[/tex].
Hence, the correct statement is:
- "Z and B are independent events because [tex]\( P(Z \mid B) = P(Z) \)[/tex]."
