To determine whether the events [tex]\(Z\)[/tex] and [tex]\(B\)[/tex] are independent, we need to compare the conditional probability [tex]\(P(Z \mid B)\)[/tex] with the marginal probability [tex]\(P(Z)\)[/tex]. If these two probabilities are equal, the events are independent. If they are not equal, the events are not independent.
Here are the steps to find these probabilities:
### Step 1: Calculate [tex]\(P(Z \mid B)\)[/tex]
The conditional probability [tex]\(P(Z \mid B)\)[/tex] is found by dividing the number of occurrences of [tex]\(Z\)[/tex] within [tex]\(B\)[/tex] by the total number of occurrences of [tex]\(B\)[/tex].
From the table:
- The number of [tex]\(Z\)[/tex] within [tex]\(B\)[/tex] ([tex]\(B \)[/tex] and [tex]\(Z\)[/tex]) is 61.
- The total number of [tex]\(B\)[/tex] (total occurrences of [tex]\(B\)[/tex]) is 151.
So,
[tex]\[
P(Z \mid B) = \frac{\text{Number of } Z \text{ within } B}{\text{Total number of } B} = \frac{61}{151} \approx 0.40397
\][/tex]
### Step 2: Calculate [tex]\(P(Z)\)[/tex]
The marginal probability [tex]\(P(Z)\)[/tex] is found by dividing the total number of occurrences of [tex]\(Z\)[/tex] by the overall total of occurrences.
From the table:
- The total number of [tex]\(Z\)[/tex] is 110.
- The overall total number is 229.
So,
[tex]\[
P(Z) = \frac{\text{Total number of } Z}{\text{Overall total}} = \frac{110}{229} \approx 0.48035
\][/tex]
### Step 3: Compare [tex]\(P(Z \mid B)\)[/tex] with [tex]\(P(Z)\)[/tex]
Now, we compare the two probabilities:
- [tex]\(P(Z \mid B) \approx 0.40397\)[/tex]
- [tex]\(P(Z) \approx 0.48035\)[/tex]
Since [tex]\(P(Z \mid B) \not= P(Z)\)[/tex], we can conclude that the events [tex]\(Z\)[/tex] and [tex]\(B\)[/tex] are not independent.
### Conclusion
The correct statement is:
[tex]\[Z \text{ and } B \text{ are not independent events because } P(Z \mid B) \neq P(Z).\][/tex]
So, the correct option is:
[tex]\[Z \text{ and } B \text{ are not independent events because } P(Z \mid B) \neq P(Z).\][/tex]
