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Let A , B and C be three events such that the probability that exactly one of A and B occurs is ( 1 - k ) , the probability that exactly one of B and C occurs is ( 1 - 2 k ) , the probability that exactly one of C and A occurs is ( 1 - k ) and the probability of all A , B and C occur simultaneously is k 2 , where 0 < k < 1 . Then the probability that at least one of A , B and C occur is:

Options

  1. Agreater than 1 8 but less than 1 4
  2. Bgreater than 1 2
  3. Cgreater than 1 4 but less than 1 2
  4. Dexactly equal to 1 2

Correct answer

B. greater than 1 2

Step-by-step solution

Given: P ( A ¯ ∩ B ) + P ( A ∩ B ¯ ) = 1 - k P ( A ¯ ∩ C ) + P ( A ∩ C ¯ ) = 1 - k P ( B ¯ ∩ C ) + P ( B ∩ C ¯ ) = 1 - 2 k P ( A ∩ B ∩ C ) = k 2 Now, P ( A ) + P ( B ) - 2 P ( A ∩ B ) = 1 - k   … i P ( B ) + P ( C ) - 2 P ( B ∩ C ) = 1 - 2 k   … ii P ( C ) + P ( A ) - 2 P ( A ∩ C ) = 1 - k   … iii Adding i ,   i i   &   i i i , we get P A + P B + P C - P A ∩ B - P

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