Question

which statement is true about whether z and b are independent events?

\\(z\\) and \\(b\\) are independent events because \\(p(z|b) = p(z)\\).
\\(z\\) and \\(b\\) are independent events because \\(p(z|b) = p(b)\\).
\\(z\\) and \\(b\\) are not independent events because \\(p(z|b) \
eq p(z)\\).
\\(z\\) and \\(b\\) are not independent events because \\(p(z|b) \
eq p(b)\\).

Explanation:

Calculate the marginal probability of Z

Using the Two-Way Frequency Tables knowledge point

$$ P(Z) = \frac{\text{Total for } Z}{\text{Grand Total}} = \frac{297}{660} = 0.45 $$

Calculate the conditional probability of Z given B

Using the Conditional Probability Calculation knowledge point

$$ P(Z|B) = \frac{\text{Frequency of } Z \text{ and } B}{\text{Total for } B} = \frac{126}{280} = 0.45 $$

Determine independence

Using the Independent Events knowledge point

$$
LATEXBLOCK0
$$

Answer:

• (A) Z and B are independent events because P(Z|B) = P(Z). (Correct answer)
• (B) Z and B are independent events because P(Z|B) = P(B).
• (C) Z and B are not independent events because P(Z|B) ≠ P(Z).
• (D) Z and B are not independent events because P(Z|B) ≠ P(B).