Posted: 1/10/2026 10:42:52 PM EDT
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How does one verify that a particular die is fair? My question is born out of seeing one of Matt Parkers' videos about mathematically determining the length of a cylinder such that it is equally likely to land on either of its ends as its side thus making a fair three sided die. The initial premise being that a long skinny cylinder (unsharpened pencil for example) is quite unlikely to land on one of its ends while a coin is unlikely to land on its edge though both can be balanced in those ways. Thus, there exists a cylinder whose length can make it land on its side or one of its ends equally likely. I never did find out if that length was ever found. Since I'm a machinist by trade, I have given the thought of finding out this length myself through trial and error. But, I'd add in a bit of complexity by putting a small, but significant, radius on the corners. I'd start with something that is intuitively slightly long, and then progressively shorten it by small increments until I found a fair D3. But, I don't know the minimum rolls required, nor the way to mathematically test the results. The bit of statistics I took in Jr. College several decades ago never got to this kind of analysis. I come here to learn and apply a "practical" problem. In addition, there are other odd shapes and number of sides which could be used for a die say D5, D7, etc. which would need to be tested. So, is there a general formula to determine the minimum number of throws to verify that the die in question is not fair, and the number throws and analysis to say that it is indeed fair? |
If I were a hedonist, I'd be a hermit.
You need a good airgun if you don't have one already.
You need a good airgun if you don't have one already.
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This problem falls under the general topic of sampling theory. There happens to be some dice geeks that go into great detail about this. There are also papers on extensive tests done in the past. You can't every be completely sure, but you can get to smaller and smaller errors in the estimate of fairness with more tries. You should have a pretty good idea with 1000 throws doing the analysis demonstrated in the above links. |
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| Since your 3 faces are not identical, your cylindrical die will be highly sensitive to the way that it's thrown. I doubt that you can create one that will be equally likely to land on any of its faces thrown from a variety of angles, speeds, heights, other initial conditions. |