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

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) ≠ p(z). z and b are not independent events because p(z | b) ≠ p(b).

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) ≠ p(z). z and b are not independent events because p(z | b) ≠ p(b).

Answer

Explanation:

Step1: Recall the independence - rule

Two events (Z) and (B) are independent if (P(Z|B)=P(Z)).

Step2: Calculate (P(Z))

The total number of outcomes is (n = 660), and the number of outcomes for event (Z) is (n_Z=297). So (P(Z)=\frac{n_Z}{n}=\frac{297}{660}=\frac{9}{20}=0.45).

Step3: Calculate (P(Z|B))

The number of outcomes for event (B) is (n_B = 280), and the number of outcomes for both (Z) and (B) is (n_{Z\cap B}=126). By the formula for conditional - probability (P(Z|B)=\frac{P(Z\cap B)}{P(B)}), and since (P(Z\cap B)=\frac{n_{Z\cap B}}{n}=\frac{126}{660}) and (P(B)=\frac{n_B}{n}=\frac{280}{660}), then (P(Z|B)=\frac{n_{Z\cap B}}{n_B}=\frac{126}{280}=\frac{9}{20}=0.45).

Answer:

Z and B are independent events because (P(Z|B)=P(Z)).