discrete math e

Question

Events A, B, and C in a sample space have P(A)=0.2, P(B)=0.4, and P(A ∪ B ∪ C)=0.9. Find P(C) if A and B are mutually exclusive, A and C are independent, and B and C are independent.

Answer 1

Since A and B are mutually exclusive, then P(A∩B)=0 and so P(A∩B∩C)=0. As A and C are independent, then P(A∩C)=P(A)*P(C). Similarly, P(B∩C)=P(B)*P(C) as B and C are independent. Now

P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C)

P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A)P(C) - P(B)P(C) + P(A∩B∩C)

0.9 = 0.2 + 0.4 + P(C) - 0 -0.2*(P(C)) - 0.4*(P(C)) + 0

0.9 = 0.6 + 0.4*P(C)

P(C) = 0.75