Assume that $n$ birds $0,1,...,n − 1$ sit around a circle. Suddenly, each bird randomly pecks the birds immediately to its left or right each with probability $1/2$. What is the expected number of unpecked birds?
I don't know if I'm thinking of this from a simple point of view, was wondering if someone can let me know if I'm correct or not.
We have n number of birds. If it pecks left or right, it's a probability of $1/2$ Unpecked would just mean they were the part of the $1/2$ probability.
So the answer would be $\textbf{(1/2)}^\textbf{n}$
Can someone let me know if my approach is correct?
Hint: Use indicator functions for each bird which are $1$ if the bird is unpecked and $0$ otherwise. Then use linearity of expectation.