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#1




Three sided coin ( basic math question)
In this PDF there is a comparison of different models describing 3sided coins.
Now,on page 3 Model 1 is pretty clear, to calculate the probability (of the flipped coin landing on its side ) they consider coin side surface area to the total surface area ratio. Model 2 is a bit confusing it seems that they divide one side of a perimeter (h) by the sum of three other sides 2(2R)+h rather than the sum of all four sides intuitively it should be 2(2R) +2 h What am I missing ? Related question: how do they define cross sectional length? Is it just a half of the diagonal, as per drawing? And if that's the case,why would the probability be proportional to that length? I don't find it very intuitive. Thanks 
#2




Sorry ,wrong forum. Intended for GQ.

#3




I didn't read the linked paper very closely yet, but this recent video by Matt Parker discusses threesided coins, or more precisely, how thick a cylinder would need to be to have a probability of exactly onethird to land on each of its ends, and on its side.

#4




Thanks, I've seen this video, but it's not what I'm asking.



#5




Moderating
Moved from GD to GQ.
[/moderating] 
#6




What four sides? There are three sides, the head face (length 2R), the tail face (length 2R) and the edge (length h).

#7




557 views and no answer ?
Am I asking a wrong question ? 
#8




Since you appear to be frustrated and a little peeved, I made the mistake of clicking the link you left and going to the PDF. My God, I've never read anything dryer or more boring!
Is this a course you have to take towards a degree? If so, my "answer" is, DROP THE COURSE!
__________________
"The greatest obstacle to discovery is not ignorance  it is the illusion of knowledge." Daniel J Boorstin 
#9




Jasmine ,
no, my question has nothing to do with courses. I was reading this and found a link to that PDF file at the end of the article. The reason I'm asking is I believe that the authors, serious mathematicians, had decided to consider model 2 on page 3. It's supposed to be a pretty basic model, but, embarrassingly, I don't follow their logic. 


#10




Quote:
I spent a few minutes wrestling with it, and specifically with the formula the OP asks about, and I don't understand it enough to come up with a useful answer. I may, or may not, come back to it later and try again when I have more time. 
#11




Quote:
Jasmine, if you don't find a question interesting, there is no need to threadshit. This kind of answer is not useful in General Questions. No warning issued, but don't do this again. Colibri General Questions Moderator 
#12




Sorry!
__________________
"The greatest obstacle to discovery is not ignorance  it is the illusion of knowledge." Daniel J Boorstin 
#13




Their derivation is wrong; you can confirm that by plugging in an eta of 2, which corresponds to a square die. You get a probability of 1/3 for a square die, which is counterfactual.
For 'cross sectional length', let's assume they mean 'perimeter along this cross section'. If they meant something else, it's sloppy that they didn't clarify. The total perimeter is 2h+2(2R) Because the two h sides are actually the same side, the probability of landing on an h based on its proportion of perimeter is : p=2h/(4R+2h) rather than h/(4R+2h) this simplifies to h/(2r+h) substitute for eta: p=eta/(2+eta) Now, if you plug in eta=2 (square crosssection), you get a probability of 1/2, which is correct. An equal probability of landing on any of four sides, but two of those sides are actually the same side: A,B,B,C rather than A,B,C,D, and we're keeping track of the probability of B. 
#14




There is technically another possibility, landing on a corner (for that matter head side corner or tail side corner). Almost zero chance of it ever happening, but not zero.



#15




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

#16




Quote:
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That doesn't sound very intuitive to me. Also, just under the Figure1 they define diagonal length l = sqrt{(R^2 +(h/2)^2)} which adds to the confusion. 
#17




Can't get to the link from my work computer, but ISTM that someone's doing an awful lot of work to come up with a fair 3sided coin when you can just use a regular 6sided die, treating opposite sides as the same outcome.

#18




Quote:
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. 
#19




Quote:
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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. Last edited by Thudlow Boink; 02282018 at 11:54 AM. 


#20




I believe our esteemed member Chronos owns/has built his very own 3sided coin [cite: some previous thread on this]. Maybe he'll show up to add his 2/3 cents.

#21




Quote:
Alternative link 
#22




Probability 0 isn’t an impossible event though.

#23




Quote:
Did some mention Toblerone lattices? Some exact results for selfavoiding random surfaces Amos Maritan and Attilio Stella The formal connection between selfavoiding 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. 
#24




Quote:
With that definition, we see that an eta of 1 (ie radius=height) gives a center crosssection 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 


#25




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.

#26




Nope, I do have a number of dice of my own design with various numbers of sides, but I've never done a d3.
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Time travels in divers paces with divers persons. As You Like It, III:ii:328 Check out my dice in the Marketplace Last edited by Chronos; 02282018 at 04:09 PM. 
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