Question

Let A and B be two events such that 0 < P(A) < 1 and 0 < P(B) < 1. Then which one of the following statements is NOT true ?

• A. If A(A|B) > P(A), then P(B|A) > P(B)
• B. If P(A ∪ B) = 1, then A and B cannot be independent
• C. If P(A|B) > P(A), then P(A^c|B) < P(A^c)
• D. If P(A|B) > P(A), then P(A^c|B^c) < P(A^c)

Answer 1

The Correct Option is D

To solve the question, we need to analyze each statement using probability principles, particularly the concept of conditional probability and independence. Let's go through each option step by step.

Option Analysis:

1. \(P(A|B) > P(A) \Rightarrow P(B|A) > P(B)\)
   • This statement uses the relationship between the conditional probabilities and the Bayes' theorem. If \(P(A|B) > P(A)\), it indicates that event A is more likely given B has occurred, thereby increasing the likelihood of B given A. Hence, \(P(B|A) > P(B)\).
2. \(P(A \cup B) = 1 \Rightarrow \text{A and B cannot be independent}\)
   • When \(P(A \cup B) = 1\), it implies that one of the events A or B must occur. For independent events, \(P(A \cup B) = P(A) + P(B) - P(A)P(B)\), but if they were independent, \(P(A)P(B) \neq 0\) and the probability would not sum to 1 unless some dependency exists. Hence, this statement is true.
3. \(P(A|B) > P(A) \Rightarrow P(A^c|B) < P(A^c)\)
   • If \(P(A|B) > P(A)\), then A is more likely to happen given B. Correspondingly, it decreases the likelihood of the complement of A under the same condition B, resulting in \(P(A^c|B) < P(A^c)\). Hence, this statement is true.
4. \(P(A|B) > P(A) \Rightarrow P(A^c|B^c) < P(A^c)\)
   • This statement is not necessarily true. \(P(A^c|B^c)\) concerns the probability of the complement of A given that B has not occurred. The occurrence likelihood of \(A^c\) given \(B^c\) does not directly follow from an increase in \(P(A|B) > P(A)\). Hence, this is the incorrect statement among the options.

Conclusion:

The statement that is NOT true is: \(P(A|B) > P(A) \Rightarrow P(A^c|B^c) < P(A^c)\). This statement incorrectly relates the conditional probabilities without direct correlation.