If P(B)= 3 5 P(A / B)= 1 2 and P(A B)= 4 5 then P(A B)^ +P (A^ B )=

Step-by-step solution

Given P(B) = 3 5 , P(A|B) = 1 2 , and P(A B) = 4 5 . Using the definition of conditional probability, P(A B) = P(A|B) P(B) = 1 2 3 5 = 3 10 . Using the addition rule P(A B) = P(A) + P(B) - P(A B) , we have 4 5 = P(A) + 3 5 - 3 10 . P(A) = 4 5 - 3 5 + 3 10 = 1 5 + 3 10 = 2+3 10 = 5 10 = 1 2 . The expression to evaluate is P(A B)' + P(A' B) . P(A B)' = 1 - P(A B) = 1 - 4 5 = 1 5 . Using the property P(A' B) = P(A' B')' = 1 - P(A' B') . By De Morgan's Law, A' B' = (A B)' , so P(A' B) = 1 - P((A B)') = 1 - (1 - P(A B))