A and B are two independent events. The probability of their simultaneous occurrence is 1 8 and the probability that neither of them occurs is 3 8 . Then their individual probabilities are

Step-by-step solution

Let P(A) = x and P(B) = y . Since A and B are independent events, P(A B) = P(A)P(B) = xy = 1 8 . The probability that neither occurs is P(A^ c B^ c ) = P(A^ c )P(B^ c ) = (1-x)(1-y) = 3 8 . Expanding the second equation: 1 - x - y + xy = 3 8 . Substituting xy = 1 8 into the equation: 1 - (x+y) + 1 8 = 3 8 . Rearranging gives x+y = 1 + 1 8 - 3 8 = 1 - 2 8 = 1 - 1 4 = 3 4 . We have the sum x+y = 3 4 and the product xy = 1 8 . These are the roots of the quadratic equation t^2 - (x+y)t + xy = 0 , which is t^2 - 3 4 t +