Three events A , B and C are such that A and B are disjoint, A and C are independent, B and C are independent. If 4P(A)=2P(B)=P(C) and P(A B C)=5P(A) , then what is the value of P(C) ?

Step-by-step solution

Let P(A) = x . From the given relation 4P(A) = 2P(B) = P(C) , we have P(B) = 2x and P(C) = 4x . Since A and B are disjoint, P(A B) = 0 and P(A B C) = 0 . Since A and C are independent, P(A C) = P(A)P(C) = x(4x) = 4x^2 . Since B and C are independent, P(B C) = P(B)P(C) = 2x(4x) = 8x^2 . Using the formula for the union of three events: P(A B C) = P(A) + P(B) + P(C) - P(A B) - P(B C) - P(C A) + P(A B C) Substituting the given values: 5x = x + 2x + 4x - 0 - 8x^2 - 4x^2 + 0 5x = 7x - 12x^2 12x^2 - 2x = 0 2x(6x - 1) = 0