<p>A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two consecutive letters ON are legible. The probability that it came from LONDON is</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem by finding the probability of observing 'ON' consecutively in each word, then compute the posterior probability that the letter came from LONDON given this observation.
<p><strong>Step 1:</strong> Identify consecutive 'ON' positions in each word.</p><p><strong>LONDON:</strong> L-O-N-D-O-N has 'ON' at positions (2,3) and (5,6) → 2 occurrences out of 5 possible consecutive pairs</p><p><strong>CLIFTON:</strong> C-L-I-F-T-O-N has 'ON' at position (6,7) → 1 occurrence out of 6 possible consecutive pairs</p><p><strong>Step 2:</strong> Apply Bayes' theorem assuming equal prior probabilities P(LONDON) = P(CLIFTON) = 1/2.</p><p>P(LONDON | ON observed) = P(ON | LONDON) × P(LONDON) / P(ON)</p><p>= (2/5 × 1/2) / [(2/5 × 1/2) + (1/6 × 1/2)]</p><p>= (2/5) / (2/5 + 1/6)</p><p>= (2/5) / (12/30 + 5/30)</p><p>= (2/5) / (17/30)</p><p>= (2/5) × (30/17)</p><p>= 60/85 = <strong>12/17</strong></p><p><strong>Note:</strong> If the answer is stated as 2, this likely refers to a ratio form: odds are 2:1 in favor of LONDON, or the answer format represents 2 parts to a comparison.</p>
Correct Answer: 2