Three sided coin ( basic math question)

[ToughLife]

@Mangetout

Me too - and I actually think the problem doesn’t have a true solution - because the asymmetry/inequality of shape of the sides means that the fairness of the thing is going to depend on how it is thrown, and other factors such as the surface onto which it falls.

I think it could quite easily be possible to design a cylindrical 3 sided coin that is ‘fair’ when dropped from 1 metre, with a 100 rpm spin, perpendicular to the axis, onto smooth glass, and find that it’s desperately unfair when dropped from a different height, at a different spin, etc.

You are right, they do take that in consideration in models 3-6.
Alternative link

[octopus]

@Thudlow_Boink

I think that, theoretically, the probability is zero, since a corner would have zero surface area.

Probability 0 isn’t an impossible event though.

[Leo_Bloom]

@Thudlow_Boink

As the link in Post #9 points out, there do exist 3-sided flippable objects.The OP’s link claims to be interested in the subject as an example of “Bayesian model comparison.”

ETA: Due to the shape and weight of the actual chocolate inside the triangular package, I suspect the three sides of a Toblerone might not all land with equal frequency.

Did some mention Toblerone lattices?

Some exact results for self-avoiding random surfaces
Amos Maritan and Attilio Stella
The formal connection between self-avoiding surfaces (SAS) and suitable lattice gauge theories is discussed in the limit that the number of field components goes to zero. Different gauge models correspond to different rules for weighting the SAS topologies or to different constraints imposed on the boundaries. The fractal dimension of a SAS model on a toblerone lattice in d = 2.58… dimensions is calculated exactly. Finally a general qualitative discussion of the behaviour of the SAS in the scaling limit is given in the light of the above and other recent results.

[brossa]

@ToughLife

if they define cross sectional length as a perimeter, why do they “presume that the probability is proportional to that length” ?
That doesn’t sound very intuitive to me.

It’s a fairly unphysical way of looking at the problem, but it’s saying ‘let’s say that a coin flip is picking a point along the perimeter at random and make a note of which side contains that point.’ Under that definition, longer sides get picked more often than short sides in proportion to their lengths.

With that definition, we see that an eta of 1 (ie radius=height) gives a center cross-section perimeter of 6R, with 2R for side A, 2R for side B, and 2R for edge. So a point drawn at random is equally likely to fall on A, B, or E

[brossa]

I should add that the first model treats the coin as a collection of surfaces, and picks a point on the surface, while the second model reduces those surfaces to line segments and picks a point along the line.

[Chronos]

Nope, I do have a number of dice of my own design with various numbers of sides, but I’ve never done a d3.