Let A and B be independent events such that P ( A ) = p , P ( B ) = 2 p . The largest value of p , for which P (exactly one of A , B occurs) = 5 9 , is:

Step-by-step solution

Probability (exactly one of A , B occurs) = 5 9 P A ∩ B ¯ + P A ¯ ∩ B = 5 9 P A - P A ∩ B + P B - P A ∩ B = 5 9 Given, A and B are independent events. So, P A ∩ B = P A P B ⇒ P ( A ) + P ( B ) - 2 P ( A ) P ( B ) = 5 9 ⇒ p + 2 p - 4 p 2 = 5 9 ⇒ 36 p 2 - 27 p + 5 = 0 ⇒ ( 12 p - 5 ) ( 3 p - 1 ) = 0 p = 1 3  or  5 12