Let A and B be two events such that the conditional probabilities P(A|B) = 2 3 and P(B|A) = 1 2 . If the probability of their union is P(A B) = 5 8 , then the conditional probability P(A'|B') is equal to

Step-by-step solution

Let P(A B) = k . Using the definition of conditional probability, we have: P(A|B) = P(A B) P(B) 2 3 = k P(B) P(B) = 3k 2 P(B|A) = P(A B) P(A) 1 2 = k P(A) P(A) = 2k By the addition theorem of probability: P(A B) = P(A) + P(B) - P(A B) 5 8 = 2k + 3k 2 - k 5 8 = 5k 2 k = 1 4 Now, we can find P(B) : P(B) = 3 2 1 4 = 3 8 The probability of the complement of B is: P(B') = 1 - P(B) = 1 - 3 8 = 5 8 Using De Morgan's laws, the probability of the intersection of the complements is: P(A' B') = P((A B)') = 1 - P(A B) P(A' B')