Let A and B be two events such that P A ∪ B ¯ = 1 6 ,   P A ∩ B = 1 4 and P A ¯ =…

Let A and B be two events such that PA∪B¯=16, PA∩B=14 and PA¯=14, where A¯ stands for the complement of the event A. Then the events A and B are 
  1. Independent but not equally likely.
  2. Independent and equally likely.
  3. Mutually exclusive and independent.
  4. Equally likely but not independent.

Solution

PA∪B¯=16         

∴    PA∪B=56

PA∩B=14

PA¯=14         

∴    PA=34

∴    PA∪B=PA+PB-PA∩B

∴    56=34+PB-14

∴    PB=13

∴    PA·PB=13×34=14=PA∩B

Therefore, A and B are independent and since PA≠PB they are not equally likely.

Asked in: JEE Main 2014 (06 Apr)

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