Let A and B are independent events with P(B)= 2 5 , P(A B)= 11 20 , then P ( A ^ B ) is root of the equation

Step-by-step solution

Given independent events A and B with P(B) = 2 5 and P(A B) = 11 20 , the independence condition P(A B) = P(A)P(B) applies. The probability of union is P(A B) = P(A) + P(B) - P(A)P(B) . Substituting values yields 11 20 = P(A) + 2 5 - 2 5 P(A) . Solving for P(A) : 11 20 - 8 20 = P(A) (1 - 2 5 ) , so 3 20 = P(A) ( 3 5 ) and P(A) = 1 4 . For independent events, P(A' B) = P(A') = 1 - P(A) = 3 4 . Testing x = 3 4 in the options shows it satisfies 4x^2 - 7x + 3 = 0 since 4 ( 9 16 ) - 7 ( 3 4 ) + 3 = 0 . Option A is corre