Say the particular individuals are X and Y and every other person is M

When you have a round table, you have to fix one person down: Let's say we fix the first M down. We are then left with (X - Y) and 10 other M's to sit down. The scheme can be represented as follows:

(X - Y) - M - M - M - M - M - M - M - M - M - M

(X - Y) can be interchanged 2! ways. It could be (X - Y) or (Y - X).

(X - Y) or (Y - X) can be seen as a single unit in and of itself just as any other M. Hence, (X -Y) along with other 10 M's can be seated 11! ways

Round table arrangement total possible outcomes: (n-1)! = (13-1)!

Unfavorable outcomes when 2 particular individuals are arranged next to each other: 2!*11!

Favorable outcomes: (12! - 2!*11!)

When we talk about probability, we have to further divide above favorable outcomes by the total outcomes: (12! - 2!*11!) / 12! = 1 - 1/6 = 5/6

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Best Regards,

Paul