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.
Best Answer
The first part is right.
For the simulation, are you also counting the case where the chosen numbers are $1$ and $N$?