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NDA 1 2024 Maths, question 113

Topic: Probability

Passage · Q113 to Q114

A,BA,B and CC are three events such that P(A)=0.6,P(B)=0.4,P(C)=0.5,P(A∪B)=0.8,P(A∩C)=0.3P(A)=0.6,P(B)=0.4,P(C)=0.5,P(A\cup B)=0.8,P(A\cap C)=0.3 and P(A∩B∩C)=0.2P(A\cap B\cap C)=0.2 and P(A∪B∪C)≥0.85P(A\cup B\cup C)\ge 0.85.

Q113

What is the minimum value of P(B∩C)P(B\cap C) ?

  1. (a) 0.10.1
  2. (b) 0.20.2Correct answer
  3. (c) 0.350.35
  4. (d) 0.450.45

Solution.

Answer: (b) 0.20.2

  1. Let t = P(B ∩ C).
  2. Since the triple intersection is contained in B ∩ C, t ≥ 0.2.
  3. Also P(A ∩ B) = 0.6 + 0.4 − 0.8 = 0.2.
  4. Inclusion–exclusion gives union probability 1.2 − t.
  5. The lower bound t = 0.2 is feasible and is the minimum.