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.