This is quite a sophisticated problem and probably the best way to solve it is to use Burnside's Lemma to count the number of essentially different necklaces.
With $3$ colours there are $3^{13}$ arrangements which can be reflected (turned over) and rotated.
The identity transformation fixes all $3^{13}$ arrangements.
The other $12$ rotations each fix just $3$ necklaces (the ones which use just one colour).
The $13$ reflections each fix $3^7$ necklaces ('opposite' beads must have the same colour).
So the number of essentially different necklaces is $$\frac{3^{13}+12\times3+13\times3^7}{26}=62415.$$
With just $2$ colours the number of essentially different necklaces is $$\frac{2^{13}+12\times2+13\times2^7}{26}=380.$$
The probability is $\frac{79}{12483}$