Class CBSE Class 12 Mathematics Probability Q #1739
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
For two events A and B such that $P(A) \ne 0$ and $P(B) \ne 1$, $P(A^{\prime}/B^{\prime})=$
(A) $1-P(A/B)$
(B) $1-P(A^{\prime}/B)$
(C) $\frac{1-P(A\cap B)}{P(B^{\prime})}$
(D) $\frac{1-P(A\cup B)}{P(B^{\prime})}$
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Correct Answer: D

AI Tutor Explanation

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Detailed Solution

Step 1: Apply the definition of Conditional Probability

The conditional probability of an event $A'$ given $B'$ is defined as: $$P(A'/B') = \frac{P(A' \cap B')}{P(B')}$$

Step 2: Apply De Morgan's Law

According to De Morgan's Law, the intersection of the complements of two sets is the complement of their union: $$A' \cap B' = (A \cup B)'$$ Therefore, $P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B)$.

Step 3: Substitute and Simplify

Substitute the result from Step 2 into the formula from Step 1: $$P(A'/B') = \frac{1 - P(A \cup B)}{P(B')}$$

Final Answer: D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to synthesize the definition of conditional probability with set-theoretic identities (De Morgan's Law) to derive a specific expression.
Knowledge Dimension: PROCEDURAL
Justification: The student must follow a sequence of logical steps involving probability axioms and set operations to reach the correct algebraic form.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the conceptual grasp of Probability (Chapter 13) beyond rote memorization, specifically focusing on the properties of complements and conditional events.

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