There are 6 people that could get their hat back, so we must multiply 6 by the number of ways that the other 5 people can arrange their hats such that no one gets his/her hat back. So, the number of ways this will happen is ( $6 \cdot$ derangement of 5 ), or $6 * 44=264$. Since there are $6!=720$ possible arrangements of hats, the probability of exactly one person getting their hat back is $\frac{264}{720}=\frac{11}{30}$.