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 exa

Question

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:

TestHub · Accepted Answer

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 B ∩ C - P C ∩ A = - 4 k + 3 2 Now, P A ∪ B ∪ C = P A + P B + P C - P A ∩ B - P B ∩ C - P C ∩ A + P A ∩ B ∩ C ⇒ P A ∪ B ∪ C = - 4 k + 3 2 + k 2 ⇒ P A ∪ B ∪ C = 2 k 2 - 4 k + 3 2 ⇒ P A ∪ B ∪ C = 2 ( k - 1 ) 2 + 1 2 ⇒ P A ∪ B ∪ C = k - 1 2 + 1 2 ⇒ P ( A ∪ B ∪ C ) > 1 2

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