If A and B are independent events such that P (A B^ )= 3 25 and P ( A ^ B )= 8 25 , then P ( A )=

Step-by-step solution

Let P(A) = x and P(B) = y , with A and B independent. Given P(A B') = 3 25 and P(A' B) = 8 25 , use independence: P(A B') = P(A)P(B') and P(A' B) = P(A')P(B) . Substituting P(B') = 1 - y and P(A') = 1 - x : x(1 - y) = 3 25 x - xy = 3 25 y(1 - x) = 8 25 y - xy = 8 25 Subtracting yields: (y - xy) - (x - xy) = 8 25 - 3 25 y - x = 1 5 y = x + 1 5 Substitute into the first equation: x(1 - (x + 1 5 )) = 3 25 x( 4 5 - x) = 3 25 Multiply both sides by 25: 25x( 4 5 - x) = 3 20x - 25x^2 = 3 Rearrange to standard form: 25x^2