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"If " `P(AcupB)=P(AcapB)`" for any two events A and B, then"`

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"If " P(A∪B)=P(A∩B) " for any two events A and B, then"A. P(A)=P(B)B. P(A)gtP(B)C. P(A)ltP(B)` D. None of these Select the correct answer from above options

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Correct Answer - A (a) Given P ( A ∪ B ) = P ( A ∩ B ) Given P ( A ) + P ( B ) − P ( A ∩ B ) = P ( A ∩ B ) P ⇒ [ P ( A ) − P ( A ∩ B ) ] + [ P ( B ) − P ( A ∩ B ) ] = 0 ⇒ But P ( A ) − P ( A ∩ B ) ≥ = 0 But and P ( B ) − P ( A ∩ B ) ≥ = 0 [ ∵ P ( A ∩ B ) ≤ P ( A ) or P ( B ) ] and ⇒ P ( A ) − P ( A ∩ B ) = 0 ⇒ and P ( B ) − P ( A ∩ B ) = 0 and [ since, sum of two non-negative numbers can be zero only when these numbers aree zero ] [ ⇒ P ( A ) = P ( A ∩ B ) ⇒ and P ( B ) = P ( A ∩ B ) and ∴ P ( A ) = P ( B ) ∴

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