In this PDF there is a comparison of different models describing 3-sided 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.
I didn’t read the linked paper very closely yet, but this recent video by Matt Parker discusses three-sided coins, or more precisely, how thick a cylinder would need to be to have a probability of exactly one-third to land on each of its ends, and on its side.
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! :smack:
Is this a course you have to take towards a degree? If so, my “answer” is, DROP THE COURSE!
**Jasmine **,
no, my question has nothing to do with courses.
I was readingthis
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.
In my limited experience, it’s pretty typical (not bad) for the type of article it is (an academic-style journal article). But it’s certainly not written for a popular audience, and it would take some time and effort to digest.
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.
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.
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 cross-section), 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.
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.
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.
Also, just under the Figure1 they define diagonal length l = sqrt{(R^2 +(h/2)^2)} which adds to the confusion.
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 3-sided coin when you can just use a regular 6-sided die, treating opposite sides as the same outcome.
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.
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.
I believe our esteemed member Chronos owns/has built his very own 3-sided coin [cite: some previous thread on this]. Maybe he’ll show up to add his 2/3 cents.
