UPSC NDA I 2018 Mathematics Solved Paper

Concept:The problem tests the understanding of basic probability formulas and conditional probability, specifically inclusion-exclusion and the definition of conditional probability.Explanation:We verify each statement separately.Statement 1:By the addition rule: $P(\overline{A} \cup B) = P(\overline{A}) + P(B) - P(\overline{A} \cap B)$.This matches the given expression, so statement 1 is correct.Statement 2:Using the addition rule for $A \cup \overline{B}$:$P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B})$.We know $P(\overline{B}) = 1 - P(B)$ and $P(A \cap \overline{B}) = P(A) - P(A \cap B)$.Substituting gives: $P(A) + 1 - P(B) - [P(A) - P(A \cap B)] = 1 - P(B) + P(A \cap B)$.This is not equal to $P(B) - P(A \cap B)$ in general, so statement 2 is incorrect.Statement 3:By definition of conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$.Rearranging gives $P(A \cap B) = P(B) \cdot P(A|B)$, which matches statement 3 exactly. Hence statement 3 is correct.Answer:Statements 1 and 3 are correct. Option B: 1 and 3 only.