<p>15 persons among whom are \(A\) and \(B\), sit down at random at a round table. The probability that there are 4 persons between \(A\) and \(B\) is ______ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: In circular arrangements, fix one person (A) to eliminate rotational symmetry, then count favorable positions for B such that exactly 4 people sit between A and B (2 on each arc). The probability is the ratio of favorable arrangements to total arrangements.
<p><strong>Step 1:</strong> For circular arrangements of n distinct objects, fix one person's position to account for rotational symmetry. Fix A at a position.</p><p><strong>Step 2:</strong> Total arrangements of 15 persons with A fixed = (15-1)! = 14! However, for probability, we count favorable vs possible positions for B relative to A.</p><p><strong>Step 3:</strong> With A fixed, there are 14 remaining positions for the other 14 persons. B can sit in any of these 14 positions with equal probability.</p><p><strong>Step 4:</strong> For exactly 4 persons between A and B at a round table: going clockwise from A, B must be at the 5th position (4 people between them), OR going counter-clockwise, B must be at the 5th position (4 people on the other side).</p><p><strong>Step 5:</strong> At a round table with A fixed, positions for B that have exactly 4 persons between A and B: The gap of 4 can occur in two ways - 4 seats clockwise or 4 seats counter-clockwise from A.</p><p><strong>Step 6:</strong> Number of favorable positions for B = 2 (one position clockwise 5 steps away, one position counter-clockwise 5 steps away)</p><p><strong>Step 7:</strong> Total possible positions for B = 14</p><p><strong>Step 8:</strong> Probability = 2/14 = 1/7 ≈ 0.1429</p><p>∴ Answer: <strong>0.1429</strong></p>
Correct Answer: 0