What is the probability that there are $k$ people between $A$ and $B$ sitting around a circle? I have $n$ people seated around a circular table and $2$ of the people are $A$ and $B$.  What is the probability that there are $k$ people between $A$ and $B$?
I have tried noting that the total possible number of ways they can all sit around the circle is $(n-1)!$ but get stuck after that.
 A: We need to define what it means for there to be $k$ people between Alicia and Beti. Because we are on a circle, we could measure distance counterclockwise, or clockwise, or let the number of people between Alicia and Beti be the smaller of the two. We choose the latter interpretation because it is marginally more complicated. 
To follow the argument below, it will be useful to draw pictures.
Imagine there are $13$ chairs. Let Alicia sit down first. Now everybody else sits down. There are $12$ equally likely choices of chair for Beti.
With probability $\frac{2}{12}$ she sits on a chair next to Alicia. Then the number of people between them is $0$.
With probability $\frac{2}{12}$ she sits on a chair one chair away from Alicia counterclockwise or clockwise. Then the number of people between them is $1$.
And so on, until finally Beti can sit six chairs away from  Alicia counterclockwise or clockwise. Again this has probability $\frac{2}{12}$, and the number of people between them is $5$.
So if random variable $X$ is the number of people between Alicia and Beti takes on the values $0$ to $5$, each with probability $\frac{2}{12}$.
If the total number of chairs is an even number, like $14$, the analyis is similar, but there is a small difference. The probability that the number of people between Alicia and Beti takes on values $0,1,2,\dots, 5$, each with probability $\frac{2}{13}$. But the probability there are $6$ people between Alicia and Beti is $\frac{1}{13}$. In this case, Beti is directly across the table from Alicia.  
A similar analysis holds in general. We must distinguish between the cases where $n$ is odd, like our $13$, or even.  One could give a "formula" in each case. The conclusion, for example, if $n$ is odd is that the number of people is $0$ to $\frac{n-3}{2}$, each with probability $\frac{2}{n-1}$. 
