Question

The probability of the simultaneous occurrence of two events A and B is p. If the probability that exactly one of A, B occurs is q, then which of the following alternatives is incorrect ? A. P( ˉ A )+P( ˉ B )=2+2q-p B. P( ˉ A )+P( ˉ B )=2-2p-q C. P(A∩B/A∪B)= p p+q D. P( ˉ A ∩ ˉ B )=1-p-q Select the correct answer from above options

Answer 1

Correct Answer - A We have, P ( A ∩ B ) = p and P ( A ) + P ( B ) + P ( B ) − 2 P ( A ∩ B ) = q P ⇒ P ( A ) + P ( B ) − 2 p + q ⇒ ⇒ P ( A ) + P ( B ) = 2 p + q ⇒ ⇒ 1 − P ( ¯¯¯ A ) + 1 − P ( ¯¯¯ B ) = 2 p + q ⇒ ⇒ P ( ¯¯¯ A ) + P ( ¯¯¯ B ) = 2 − 2 p − q ⇒ So, alternative (b) is correct. Now, { ( A ∩ B ) / ( A ∪ B ) } = P [ ( A ∩ B ) ∩ ( A ∪ B ) ] P ( A ∪ B ) { ⇒ P { ( A ∩ B ) / ( A ∪ B ) } = P ( A ∩ B ) P ( A ∪ B ) ⇒ ⇒ P { ( A ∩ B ) / ( A ∪ B ) } = P ( A ∩ B ) P ( A ) + P ( B ) − P ( A ∩ B ) ⇒ ⇒ P { ( A ∩ B ) / ( A ∪ B ) } = p 2 p + q − p = p p + q ⇒ So, alternative (c ) is correct. Finally, P ( ¯¯¯ A ∩ ¯¯¯ B ) = P ( ¯¯¯¯¯¯¯¯¯¯ A ∪ B ) = 1 − P ( A ∪ B ) P ⇒ P ( ¯¯¯ A ∩ ¯¯¯ B ) = P ( ¯¯¯¯¯¯¯¯¯¯ A ∪ B ) = 1 − [ P ( A ) + P ( B ) − P ( A ∩ B ) ] ⇒ ⇒ P ( ¯¯¯ A ∩ ¯¯¯ B ) = P ( ¯¯¯¯¯¯¯¯¯¯ A ∪ B ) = 1 − [ 2 p + q − p ] = 1 − p − q ⇒ So, alternative (d) is correct. Hence, alternative (a) is incorrect.