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NDA 1 2025 Maths, question 116

Topic: Probability

Q116

If P(A)=1/3P(A) = 1/3, P(B)=1/2P(B) = 1/2 and P(A∩B)=1/4P(A \cap B) = 1/4, then what is the value of P(Ac∩Bc)P\left( A^{c} \cap B^{c} \right)?

  1. (a) 1/4
  2. (b) 5/12Correct answer
  3. (c) 7/12
  4. (d) 11/12

Solution.

Answer: (b) 5/12

  1. By De Morgan’s law, Ac∩Bc=(A∪B)cA^{c} \cap B^{c} = (A \cup B)^{c}.
  2. Hence the probability is 1−[P(A)+P(B)−P(A∩B)]1 - \left\lbrack P(A) + P(B) - P(A \cap B) \right\rbrack.
  3. This equals 1−(1/3+1/2−1/4)=1−7/12=5/121 - (1/3 + 1/2 - 1/4) = 1 - 7/12 = 5/12.