The odds in favour of an event A are 1:1 and the odds against an event B are 2:3 . If the probability that neither A nor B occurs is 0.3 , then which of the following quadratic equations has roots P(A B ) and P(B A ) ?

Step-by-step solution

Odds in favour of A are 1:1 P(A) = 1 1+1 = 0.5 Odds against B are 2:3 P(B) = 3 2+3 = 0.6 Probability that neither A nor B occurs is 0.3 P( A B ) = 0.3 By De Morgan's laws, P( A B ) = 0.3 P(A B) = 1 - 0.3 = 0.7 Using the addition theorem, P(A B) = P(A) + P(B) - P(A B) = 0.5 + 0.6 - 0.7 = 0.4 Now, P(A B ) = P(A B ) P( B ) = P(A) - P(A B) 1 - P(B) = 0.5 - 0.4 1 - 0.6 = 0.1 0.4 = 1 4 Similarly, P(B A ) = P(B A ) P( A ) = P(B) - P(A B) 1 - P(A) = 0.6 - 0.4 1 - 0.5 = 0.2 0.5 = 2 5 The sum of the roots is 1 4 + 2 5 = 13 2