Help w/ problem, N people occupy N points in a circle and move randomly

Finnegan

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Suppose we have n points in a circle with 1 person at each point. If every person randomly moves either left or right to one of their direct neighbors, what is the expected number of empty points in the circle after everyone moves?


I'm having some trouble figuring out exactly how to go about solving this problem. My current thought process is that the neighbors of every person have to make a certain choice for a point to be empty, but I can't put this into a formula. I feel that indicator random variables could help.
 
Suppose we have n points in a circle with 1 person at each point. If every person randomly moves either left or right to one of their direct neighbors, what is the expected number of empty points in the circle after everyone moves?


I'm having some trouble figuring out exactly how to go about solving this problem. My current thought process is that the neighbors of every person have to make a certain choice for a point to be empty, but I can't put this into a formula. I feel that indicator random variables could help.
n points in a circle with 1 person at each point

Thus there was no empty seat - so there will be none left...
 
n points in a circle with 1 person at each point

Thus there was no empty seat - so there will be none left...
No, it's not that easy. Suppose n= 3 and the people are "A", "B", and "C". Suppose A and B both move to their left while C moves to the right. We end up with "B", "AC" (both A and C occupying the same point), and one empty space.
 
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