Answer: 2/5

## Explanation This is a conditional probability problem that can be solved using Bayes' Theorem. ### Step 1: Define Events - Let **A** = "Wife received the letter" - Let **B** = "Man does not receive a response" ### Step 2: Known Probabilities - P(A) = 2/3 (probability wife receives letter) - P(Aᶜ) = 1/3 (probability wife does not receive letter) - If wife receives letter, she will certainly mail a response - The response letter has the same delivery probability: P(response delivered | wife received letter) = 2/3 ### Step 3: Calculate Conditional Probabilities - P(B|A) = Probability no response given wife received letter = 1 - 2/3 = 1/3 (Since if wife receives letter, she sends response, but response may not be delivered) - P(B|Aᶜ) = Probability no response given wife did not receive letter = 1 (If wife never received letter, she cannot send response) ### Step 4: Apply Bayes' Theorem We want P(A|B) = Probability wife received letter given no response Using Bayes' Theorem: P(A|B) = [P(B|A) × P(A)] / [P(B|A) × P(A) + P(B|Aᶜ) × P(Aᶜ)] P(A|B) = [(1/3) × (2/3)] / [(1/3) × (2/3) + (1) × (1/3)] P(A|B) = (2/9) / (2/9 + 1/3) P(A|B) = (2/9) / (2/9 + 3/9) P(A|B) = (2/9) / (5/9) P(A|B) = 2/5 ### Step 5: Interpretation Given that the man did not receive a response, there is a 2/5 probability that his wife actually received his letter. This makes sense because even if she received it, the response letter might not have been delivered.

Author: LeetQuiz .