# 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.

• I guess you could inverse the problem by saying: I have a galette cut in 12 evenly sized slices (30° each). Now I throw a small ball with radius $r_f$ on that galette and I am wondering with which probability it hits a cut line. This is an alterantive problem to drawing evenly distanced lines with distance $d$ on a piece of paper and throwing a needle with length $d'<d$ on there, looking for the probability that it has an intersection with a line when it lands... see here: en.wikipedia.org/wiki/Buffon%27s_needle_problem Jan 3, 2022 at 19:51
• It should be said that the term "rois" (kings) do not refer to the kings of France but to the three Magi. Jan 3, 2022 at 19:52
• To prevent a misunderstanding which I might have fallen vicitim to: Do you only want to cut once? Or do you want to cut several times to redistribute the galette? Jan 3, 2022 at 19:54
• A solution (in french): zestedesavoir.com/articles/3409/… Jan 3, 2022 at 19:55
• @JeanMarie: thank you! everything has already been asked (and answered) I see :)
– WoJ
Jan 3, 2022 at 20:00

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 $$\frac{\left(\arcsin \frac{r_f}{d_f}\right) r_g^2}{\pi r_g^2}=\frac{\arcsin \frac{r_f}{d_f}}{\pi}$$
• Am I missing something or shouldn't you divide the area of the circular sector by $2$? Jan 3, 2022 at 20:58
• Thank you. I did not make it clear in my question that the placement of the fève is random (I probably made it worse by explicitly mentioning $d_f$ and not highlighting enough its distribution. I will clarify the question.