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Class 12 Probability · Long Answer3 Marksindependence

Class 12 Maths · Probability · Long Answer #53

Independent A,B satisfy P(A∩B′)=1/4 and P(A′∩B)=1/6. Find all possible (P(A),P(B)).

✅ Step-by-Step Solution

Final Answer: (3/4,2/3) or (1/3,1/4)
  1. Write x=P(A), y=P(B). Independence gives x(1−y)=1/4 and (1−x)y=1/6.
  2. The first equation implies y=1−1/(4x), with x>0. Substitute in the second: (1−x)(1−1/(4x))=1/6.
  3. Simplify to 12x²−13x+3=(4x−3)(3x−1)=0. Thus x=3/4 or 1/3, giving y=2/3 or 1/4 respectively.
  4. Both pairs lie in [0,1]² and satisfy the original equations: (3/4,2/3) and (1/3,1/4).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Write x=P(A), y=P(B). Independence gives x(1−y)=1/4 and (1−x)y=1/6.
  • [1 mark] The first equation implies y=1−1/(4x), with x>0. Substitute in the second: (1−x)(1−1/(4x))=1/6.
  • [1 mark] Simplify to 12x²−13x+3=(4x−3)(3x−1)=0. Thus x=3/4 or 1/3, giving y=2/3 or 1/4 respectively.
  • [0.5 marks] Both pairs lie in [0,1]² and satisfy the original equations: (3/4,2/3) and (1/3,1/4).

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