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NDA 1 2024 Maths, question 114

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.

Q114

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

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

Solution.

Answer: (c) 0.350.35

  1. Inclusion–exclusion gives P(A ∪ B ∪ C) = 1.2 − t, where t = P(B ∩ C).
  2. The condition 1.2 − t ≥ 0.85 yields t ≤ 0.35.
  3. This is feasible: the seven internal Venn regions can be assigned A-only 0.3, B-only 0.05, C-only 0.05, AB-only 0, AC-only 0.1, BC-only 0.15, and triple 0.2.
  4. Thus the maximum is 0.35.