Respuesta :
To find ( P(A' / B') ), which represents the probability of the complement of event A given the complement of event B, we use the formula for conditional probability:[ P(A' / B') = \frac{P(A' \cap B')}{P(B')} ]First, we need to find ( P(A' \cap B') ), the probability of the intersection of the complements of events A and B.Since ( A' ) is the complement of event A, ( P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} ).Similarly, ( B' ) is the complement of event B, so ( P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} ).Given that ( P(A \cap B) = \frac{1}{4} ), we can find ( P(A' \cap B') ) using the complement rule:[ P(A' \cap B') = 1 - P(A \cup B) ][ P(A' \cap B') = 1 - \left( P(A) + P(B) - P(A \cap B) \right) ][ P(A' \cap B') = 1 - \left( \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \right) ][ P(A' \cap B') = 1 - \left( \frac{6}{12} + \frac{4}{12} - \frac{3}{12} \right) ][ P(A' \cap B') = 1 - \frac{7}{12} = \frac{5}{12} ]Now, we can find ( P(A' / B') ):[ P(A' / B') = \frac{P(A' \cap B')}{P(B')} = \frac{\frac{5}{12}}{\frac{2}{3}} ][ P(A' / B') = \frac{5}{12} \times \frac{3}{2} = \frac{5}{8} ]So, ( P(A' / B') = \frac{5}{8} ), which corresponds to option (C).
