# Probability that a given pair of people sits next to each other in a round table.

There's a round table with N chairs. $$N$$ people are seated randomly. What is the probability that a given pair of people will sit next to each other.

If you fix the first person to sit at the i-th chair, then it seems the problem reduces down to the probability of the second person sitting in the i+1 or i-1 chair locations, which is simply just $$\frac{2}{N-1}$$. Is this correct?

When I try to prob this with monte carlo, I seem to be getting a consistently smaller probability. The way my monte carlo algorithm works is I sample 2 numbers without replacement from $$\sim U(1, N)$$. Then I check that the absolute difference between the numbers is 1. If it is, then they're deemed to be sitting next to each other. Then I divide this amount by the number of trials.

• You probably discard the pair $(1,N)$ in the simulation. – user Feb 28 at 21:54
• @user Ah you're right! – user5965026 Mar 1 at 0:31

For the simulation, are you also counting the case where the chosen numbers are $$1$$ and $$N$$?