Let A and B be two events such that P(A / B)= 1 2 and P(B / A)= 1 3 and P(A B)= 1 6 then, which one of the following is not true?

Step-by-step solution

Given P(A|B) = P(A B) P(B) = 1 2 and P(A B) = 1 6 , we have 1/6 P(B) = 1 2 , which implies P(B) = 1 3 . Given P(B|A) = P(A B) P(A) = 1 3 and P(A B) = 1 6 , we have 1/6 P(A) = 1 3 , which implies P(A) = 1 2 . Check for independence: P(A) P(B) = 1 2 1 3 = 1 6 . Since P(A B) = 1 6 , we have P(A B) = P(A)P(B) , so A and B are independent events. Evaluate P(A' B) = P(B) - P(A B) = 1 3 - 1 6 = 1 6 . This is true. Evaluate P(A B) = P(A) + P(B) - P(A B) = 1 2 + 1 3 - 1 6 = 3+2-1 6 = 4 6 = 2 3 . This is true. Since A and B