Discussion
If \(P(A | B) = P(B)\), then :
Answer: D
The answer is \(P^{2}(B) \le P(A)\).
\(P(A | B) = P(A \cap B)/P(B) = P(B)\) gives \(P(A \cap B) = P^{2}(B)\). Since \(A \cap B \subseteq A\),
\[ P^{2}(B) = P(A \cap B) \le P(A) \]
The others need not hold. Take \(P(A) = 0.7\), \(P(B) = 0.4\) and \(P(A \cap B) = 0.16\), which is consistent and gives \(P(A | B) = 0.4 = P(B)\). Then \(P(B | A) = 0.16/0.7 \approx 0.23 \ne P(A)\); A and B are not independent, since \(P(A)P(B) = 0.28 \ne 0.16\); and \(P^{2}(A) = 0.49 \gt P(B)\).
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