Cutting a galette and hitting the fève

We have in France a tradition of eating in January countless galettes des rois^(*). Hidden inside is a fève, a small figurine (it was originally a coin). The one who gets the fève without breaking a tooth is crowned queen or king.

To give some context, the home-made galette we ate today, together with its fève

As I was cutting the galette, my son asked

I wonder what the probability to hit the fève when making a cut is?

Now I wonder as well.

In the tradition of spherical cows in a vacuum, a galette with its fève can be simplified as

where $r_g$ and $r_f$ are the radii of, respectively, the galette and the fève. $d_f$ is the distance of the center of the fève from the center of the galette. EDIT: the placement of the fève is random.

Asking for a full calculation of the probability would be too much :), so my question is: how should I approach this calculation, especially since it will be dependent on $d_f$ (which will probably have a squared distribution). Any hints and warnings are welcome^(**).

^(*) We are of course talking about the only proper one - the northern one (in case someone has doubts from Wikipedia). The proper drink for a galette des rois is cidre, of course from Brittany

^(**) The prize could be a part of the galette but it is already gone.

J'adore la galette des rois.

Using your diagram, and assuming a cut with the knife is a a full line segment from the center of the galette to the edge of the crust, the probability of hitting the feve is ratio of the green area including the red area) over the area of the full disk.

Since $\widehat {OBA}$ is a right triangle, you can compute the angle at $O$ as $$\hat O = 2\arcsin \frac{BA}{OA} = 2\arcsin \frac{r_f}{d_f}$$ Thus the probability is $$\frac{\left(\arcsin \frac{r_f}{d_f}\right) r_g^2}{\pi r_g^2}=\frac{\arcsin \frac{r_f}{d_f}}{\pi}$$