Will the two of them sit directly across?

Question

Six people sit down in random order at a round table with 6 evenly spaced seats. Two of them, Priya and Sam, are hoping to sit directly across from each other.

What is the probability Priya and Sam end up in seats directly opposite one another?

Accepted Answer

Fix Priya's seat as a reference point, on a round table only relative positions matter. Once Priya is seated, Sam is equally likely to take any of the 5 remaining seats. Exactly one of those 5 seats is directly across from Priya. So P(	ext{opposite}) = \frac{1}{5} = 0.2. Takeaway. In circular seating, anchor one person and measure everyone else relative to them, this removes the rotational symmetry that otherwise inflates the count. The other person is then uniform over the remaining seats, and you just count the favorable ones.

Will the two of them sit directly across?

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