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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

enter image description here enter image description here

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

enter image description here

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.

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  • $\begingroup$ 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 $\endgroup$
    – Jfischer
    Jan 3, 2022 at 19:51
  • $\begingroup$ It should be said that the term "rois" (kings) do not refer to the kings of France but to the three Magi. $\endgroup$
    – Jean Marie
    Jan 3, 2022 at 19:52
  • $\begingroup$ 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? $\endgroup$
    – Jfischer
    Jan 3, 2022 at 19:54
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    $\begingroup$ A solution (in french): zestedesavoir.com/articles/3409/… $\endgroup$
    – Jean Marie
    Jan 3, 2022 at 19:55
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    $\begingroup$ @JeanMarie: thank you! everything has already been asked (and answered) I see :) $\endgroup$
    – WoJ
    Jan 3, 2022 at 20:00

1 Answer 1

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J'adore la galette des rois.

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 $$\frac{\left(\arcsin \frac{r_f}{d_f}\right) r_g^2}{\pi r_g^2}=\frac{\arcsin \frac{r_f}{d_f}}{\pi}$$

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  • $\begingroup$ Am I missing something or shouldn't you divide the area of the circular sector by $2$? $\endgroup$ Jan 3, 2022 at 20:58
  • $\begingroup$ Indeed, you are correct, sir. $\endgroup$ Jan 3, 2022 at 21:53
  • $\begingroup$ 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. $\endgroup$
    – WoJ
    Jan 4, 2022 at 8:56

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