Thanks for pointing out. We've re-framed the question and provided the explanation as the image below. Let: - **A** : the wife received the letter - **Ā** : the wife did not receive the letter - **B** : the man received a response letter from his wife - **B̄** : the man did not receive a response letter from his wife The question “If the man does not receive a response letter from his wife, what is the probability that his wife received his letter?” is equivalent to finding **P(A | B̄)**. By Bayes' theorem: $$ P(A \mid \overline{B}) = \frac{P(A \cap \overline{B})}{P(\overline{B})} = \frac{P(\overline{B} \mid A) \cdot P(A)}{P(\overline{B})} $$ And $$ P(\overline{B}) = P(\overline{B} \mid A) \cdot P(A) + P(\overline{B} \mid \overline{A}) \cdot P(\overline{A}) $$ Given values (from the problem context): - P(A) = ²/₃ - P(Ā) = ¹/₃ - P(B̄ | A) = ¹/₃ - P(B̄ | Ā) = 1 Substitute: $$ P(\overline{B}) = \left( \frac{1}{3} \times \frac{2}{3} \right) + \left( 1 \times \frac{1}{3} \right) = \frac{2}{9} + \frac{3}{9} = \frac{5}{9} $$ $$ P(A \mid \overline{B}) = \frac{ \frac{1}{3} \times \frac{2}{3} }{ \frac{5}{9} } = \frac{ \frac{2}{9} }{ \frac{5}{9} } = \frac{2}{5} $$ **Final answer: ²/₅**