Amar, Akbar, and Anthony write a test independently. The probability that Amar passes the test and Akbar fails the test is 3 20 . The probability that Akbar passes the test and Anthony fails the test is 1 4 . The probability that Amar and Anthony both pass the test is 2 5 . Then the probability that at least one of Amar, Akbar and Anthony fails the test is

Step-by-step solution

Let the probabilities of Amar, Akbar, and Anthony passing the test be x , y , and z respectively. Since the tests are written independently, the events are independent. We are given: x(1 - y) = 3 20 y(1 - z) = 1 4 xz = 2 5 From the third equation, z = 2 5x . Substituting this into the second equation gives: y (1 - 2 5x ) = 1 4 y ( 5x - 2 5x ) = 1 4 y = 5x 4(5x - 2) Substituting y into the first equation gives: x (1 - 5x 4(5x - 2) ) = 3 20 x ( 20x - 8 - 5x 4(5x - 2) ) = 3 20 15x^2 - 8x 20x - 8 = 3 20 20(15x^2 - 8x)