Rainbow Ordering

Question

The $7$ letters $ROYGBIV$ are permuted such that all permutations are equally likely. We say that two strings $a_1a_2\dots a_n$ and $b_1b_2\dots b_n$ are equal up to rotation if there exists some integer $0 \leq i \leq n-1$ such that $a_1a_2\dots a_n = b_{i_1}b_{i_2}\dots b_{i_n}$, where $i_k = 1 + (i + k) \hspace{3pt} \text{mod} \hspace{3pt} n$. In other words, they are the same strings up to some shift. Find the probability that $ROYGBIV$ and generated permutation are equal up to rotation.