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JEE Main2014MathematicsProbabilityActual

Let A and B be two events such that P A ∪ B ¯ = 1 6 , P A ∩ B = 1 4 and P A ¯ = 1 4 , where A ¯ stands for the complement of the event A . Then the events A and B are

Options

  1. AIndependent but not equally likely.
  2. BIndependent and equally likely.
  3. CMutually exclusive and independent.
  4. DEqually likely but not independent.

Correct answer

A. Independent but not equally likely.

Step-by-step solution

P A ∪ B ¯ = 1 6 ∴     P A ∪ B = 5 6 P A ∩ B = 1 4 P A ¯ = 1 4 ∴     P A = 3 4 ∴     P A ∪ B = P A + P B - P A ∩ B ∴     5 6 = 3 4 + P B - 1 4 ∴     P B = 1 3 ∴     P A · P B = 1 3 × 3 4 = 1 4 = P A ∩ B Therefore, A and B are independent and since P A ≠ P B they are not equally likely.

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