which two events are independent? a and x a and y b and x b and y

Explanation:

Step1: Recall independence formula

Two events (E) and (F) are independent if (P(E\cap F)=P(E)\times P(F)). In terms of the contingency - table, if (E) and (F) are events, (P(E)=\frac{\text{Total of row/column of }E}{\text{Grand total}}), (P(F)=\frac{\text{Total of row/column of }F}{\text{Grand total}}), and (P(E\cap F)=\frac{\text{Cell value of intersection}}{\text{Grand total}}).

Step2: Check for (A) and (X)

(P(A)=\frac{30}{100} = 0.3), (P(X)=\frac{50}{100}=0.5), (P(A\cap X)=\frac{15}{100} = 0.15). And (P(A)\times P(X)=0.3\times0.5 = 0.15=P(A\cap X)).

Step3: Check for (A) and (Y)

(P(A)=\frac{30}{100}=0.3), (P(Y)=\frac{28}{100} = 0.28), (P(A\cap Y)=\frac{5}{100}=0.05). And (P(A)\times P(Y)=0.3\times0.28 = 0.084\neq0.05).

Step4: Check for (B) and (X)

(P(B)=\frac{20}{100}=0.2), (P(X)=\frac{50}{100}=0.5), (P(B\cap X)=\frac{5}{100}=0.05). And (P(B)\times P(X)=0.2\times0.5 = 0.1\neq0.05).

Step5: Check for (B) and (Y)

(P(B)=\frac{20}{100}=0.2), (P(Y)=\frac{28}{100}=0.28), (P(B\cap Y)=\frac{8}{100}=0.08). And (P(B)\times P(Y)=0.2\times0.28 = 0.056\neq0.08).

Answer:

A. A and X