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I intend to get a bit closer to my goal of finding a perfectly fair die. What I mean by that is a die that has as little as possible imperfections that might cause it to favor any particular side, or in other words has as little bias as possible.

Definition of Fairness and Bias

Each side of a die with equally sized and positioned faces theoretically has an equal probability of facing up after rolling. Taking a six-sided die as an example, that would be a chance of 1/6 for each face. If you would roll this die an infinite amount of times, each face should come up evenly - if you "only" roll it a large number of times, it should roughly be the same. This is stated by the strong law of large numbers (related to the convergence of random variables).

If that is the case, you have a fair die, meaning a die that fits your expectation that each side has a 1 in 6 chance of facing up. If that is not the case, for example because one number comes up significantly fewer or more often, you have a biased die.

Definition of Die/Dice

I'm interested in information/suggestions about any kind of physical die. This means that other dice than a six-sided die are welcome, however they do have to be dice by the definition of "throwable objects with [distinguishable] sides that can rest in multiple positions".

Concerns

Any die in the physical world will, through a multitude of factors (such as their own shape, material, or environmental conditions like the rolling surface or air resistance), never perfectly reach that average. This question is about finding dice with properties that best eliminate any elements that introduce imbalance in the die itself, such as uneven weight, favoring rolling behavior, and so on.

More specifically, the following practical questions come to mind:

  • What kind of dice should I buy in the first place?
    This relates to material, manufacturer, identifiable general qualities, or even whether self-cast dice are better than commercially acquirable ones.
  • Is there anything I can/should do to those dice either before or after rolling them?
  • How can I test whether my dice are actually balanced?
  • Are there any other factors I need to consider?

Additional Notes

I am in particular interested in dice I can use to be as fair as possible when I myself want them to be fair, as opposed to dice or methods that are manufactured in order to prevent cheating, unless the first statement is also true.

For completeness's sake, while playing games with friends, most dice will be sufficient to have fun, as you should never unintentionally end up with dice that always roll or significantly favor a certain number. This question is about practical application, but more for the maniac than the otherwise focussed player.
Because this has come up a number of times, I will explicitly state that I am not advertising that only perfectly fair dice are good dice, or that you can't have fun without them.

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    If fairness is the only concern, use a random number generator. Aug 1 at 13:53
  • @ArcanistLupus dice in fact are random number generators, but assuming you mean computer programs that take care of that job, it isn't trivial to generate good pseudo-randomness either. In any case, the question is specifically about physical dice. Aug 1 at 17:44
  • I'm confused by what you mean by fairness and balance. Are you suggesting that after say 6 rolls of a d6 each number should have appeared once? What are you wanting this for where their perception of fairness is so important as opposed to a fun boardgames? Aug 1 at 18:00
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    Granted that you are using established mathematical definitions for "fairness" and "bias", I think it would improve the question to add short summaries of the definitions, to account for the fact that this site is not focused on mathematics or statistics and many readers may not be familiar with the specific meanings you are referring to.
    – murgatroid99
    Aug 1 at 21:43
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    Take some casino die. If you find out they are imbalanced, do another visit to said casino.
    – Zibelas
    Aug 2 at 7:29

2 Answers 2

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Theory

Fairness of dice is a sometimes loose concept. While there are games in which each result matters significantly (i.e. Yahtzee), there are games like pen and paper RPGs where there's a difficulty rating, and whether you succeed by rolling a 16 or 17 isn't important. Of course, a different check might succeed on a 17, but fail on a 16, but what I'm trying to get at is that not in all situations, perfect fairness matters. But of course, a perfectly fair die is never worse than a biased one in this regard.

Another thing I want to note is that while the question is about the die as an object itself, how it's being tossed is another matter entirely, which influences how fairly a die will actually roll, regardless of its physical capabilities of doing so, and even if you're not actively trying to cheat on the toss.

Manufacturing

As far as my research goes, accuracy is mainly attempted to be achieved at the production stage, so starting out with well-manufactured dice seems to be mandatory. This consists of getting a mold free of air bubbles, avoid bias from finishing the die (such as from tumbling). and mitigate the uneven amount of material taken out of the die when carving numbers into them.

Six-Sided Dice
If you're looking for six-sided dice, there's two great commonly available options: Casino dice, and precision Backgammon dice. Both are made with accuracy in mind, and should offer the highest amount of fairness you can wish to acquire due to the stakes involved in using them (referring to casino dice in particular).

Non-Six-Sided Dice
If you're looking for less common dice, such as the typical d4, d6, d8, d10, d12, d20, d100 assortment of pen&paper dice, the ones I've found to be well-manufactured are GameScience/Zocchi precision dice. There's this test that basically concludes that non-precision Chessex dice are good enough, however also shows that precision dice have a more even spread, which is what this question is asking for. There's a video by Louis Zocchi, explaining the reasons for that, basically stating that tumbling destroys the balance of the die.
I personally own four sets of these dice, inked them with crayons (which should mitigate the missing material more than a pen would, but I cannot provide any measurements in terms of how well this works, or even whether I potentially made it worse in terms of balance).
I believe I've come across precision dice from another company over the years, but never owned a set of them, and I failed to find them when looking for it now.

Opaque vs. Transparent Dice
Air bubbles may form within a die during the process of injection molding, which may result in an unbalanced die. That can be avoided (in part or full) by injecting the material more slowly, as well as letting the dice cure (cool) for a longer amount of time, which is done for transparent dice, where air bubbles would otherwise be visible. For opaque dice, this slower process isn't necessary to achieve a perfect appearance, as you cannot spot any air bubbles forming under the surface, so it makes sense for manufacturing efficiency reasons to speed up the process, resulting in the aforementioned air bubbles. This has apparently been confirmed by dice manufacturers, and makes sense if you consider that outside of this question thread, "good enough" works for most players. The previous link states that this might not actually affect the balance of the dice, which of course is true (especially considering that dice manufacturers may use different processes), but tests exist that at least some opaque dice are far more biased than transparent dice (although I will point out that this is certainly not a Chessex-specific issue - it's just what the person in the video happened to test).

Shape

Persi Diaconis introduced me to the concept of fairness by symmetry through his YouTube videos, going back to Euclid and Archimedes. The idea is basically that a perfectly symmetric die (which can be a d4, d6, d8, d12 or d20) is more fair than a non-perfectly-symmetric one, even if the latter features sides of equal shapes and dimensions. If I paraphrase him correctly, that is because of the ability to manipulate a non-perfectly-symmetric die more easily through use of manual dexterity, as well as a higher difficulty of predicting the "random" outcome for symmetric dice, even though the randomness in a die toss is technically only determined by physics and therefore in theory computeable. The concept is perhaps more of a philosophical than practical one, as your definition of fairness may differ from his, but I think it's worth mentioning in any case.

Evaluation

Whether you want to check if your precision dice are as accurate as they claim to be, or you want to test whether a set of non-precision dice is accurate enough, having a way to determine it for any particular dice is handy. There are three methods I know of to evaluate dice:

Rolling them a large number of times, and see how evenly spread the results are
This method is likely to give you a very good idea of the fairness of your dice, as long as you are willing to roll a ridiculous amount of times, and go through the effort to document every roll. Of course, keep in mind that this still is subject to probability, so you're not very likely to actually get even results, even for a perfectly balanced die. The test I already linked above should give a good idea of how a "good" result will look in terms of variance. In addition, Persi Diaconis states that "The notion of long-term frequency [is] actually pretty fictional" (source).

The Chi-squared test
The Chi-squared test is a more sophisticated method of analyzing dice rolls. Instead of leaving the interpretation of the results to you, it provides a formula to calculate whether your die's results have equal frequencies or not, based on the null hypothesis. It involves slightly more work as the previous one, as in addition to a sufficient amount of samples, you need to perform calculations with those numbers. However there are programs you can use to trivialize this task, although I dare not pick a favorite to link to at this point in time.

Spinning a die in salted water
This method takes the roll out of the equation and basically tests the balance of the die without any need for comparing results, making it a quick alternative to the above method. If your die appears to be spinning randomly and keeps showing different results, there's a very good chance that it's rather fair. I keep phrasing this rather ambiguously, because if your die has an even distribution between the majority of the numbers, but will roll a few of them more or less often than it should, there's a good chance you'll miss it using this method.

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  • Do you know if metal dice are more or less balanced than plastic ones? Aug 1 at 13:55
  • @ArcanistLupus unfortunately, I don't have experience with metal dice, since they're not what I like to play with. I'm not disqualifying them as an answer or anything, I just cannot provide any kind of experience on the matter... Hopefully, other answers can sooner or later grant more insight to that aspect. Aug 1 at 17:46
  • "The downside of this method is that it involves more work as the previous one, as you need a sufficient amount of samples" - chi-squared merely lets you appropriately analyze the data you have. It doesn't require more samples, a non-statistical analysis based on a small sample size can only be more flawed than a chi-squared test on that same sample - if you didn't have enough samples to run a useful chi-squared test, you didn't have enough to do non-statistical inference anyway. 99% of the effort here is collecting the data, plugging 6 numbers into a chi-squared calculator is trivial. Aug 1 at 18:32
  • @NuclearHoagie I merely stated that it is more work, not that the amount of work isn't worth it. As a matter of fact, I ordered the two tests the way I did because I meant to illustrate how the Chi-squared test can get you a better interpretation of the same samples. I disagree with your statement that filling the values in the calculator is trivial, because to do so, you first need to familiarize yourself with how the process works in general, but assuming that has already happened, you are of course corrent. I'll edit my answer in the attempt to reflect that. Aug 1 at 20:47
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Consider not use dice but dice cards. Such as these designed for Catan.

There are 36 cards where 2 appears once, 3 twice, and so on up to 7 appearing 6 times and back down to 12 once. This echos the distribution of 2D6.

Before the game these are shuffled as normal and a special reshuffle-card is inserted in the bottom 5. During play when a dice would be rolled you turn over the top card and evaluate the result. when you the reach the reshuffle-card the discards are shuffled back into the deck and reshuffled again with special card again added back to the bottom 5.

This means in a game you are far less likely to get an anomaly where 6 is being rolled much more than an 8. Using dice cards you will mimic close to a standard distribution over 2D6 but not in a way that is totally predictable due to way last few cards in a deck might not get used.

I'm aware this isn't a dice as I believe randomness is random and the dice provided with any game are sufficient to provide fun.

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    Balance and fairness of a die aren't subjective; they're measureable as indicated by statistical mathematics. What is subjective is whether the importance of fair and balanced dice is important, and that is not what the question is about. Aug 1 at 21:01
  • While it seems the OP dislike this answer, I think it does adds good information. In particular: "How can I test whether my dice are actually balanced?" --> those cards are actually balanced. "Are there any other factors I need to consider?" --> yes, instead of a die, you can use cards that are actually balanced.
    – Cohensius
    Aug 2 at 9:54
  • @Cohensius I agree that this post contains useful information in part, for someone. I'm glad that there's great substitutions for dice to solve their respective shortcomings, however this thread is about "how to acquire unbiased dice", not "how to replace dice with a better method". I'm sure if a casino were to approach StartPlayer and ask them about dice, they wouldn't suggest to use cards instead for their Craps game. Aug 2 at 10:43
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    Use of the "reshuffle-card" makes it highly non-random - as non-random as, say, 1-deck blackjack (which no casino will play, as counting is way too effective a strategy). This would be successful (until the first card is marked, I guess) if it was 100% "picked with replacement". The reshuffle-card game is trying to make things "look fair" more than actually being fair. It is a good solution to the "unbiased results" problem (if done right); as poor a solution to "get most unbiased dice" as random number apps.
    – Mycroft
    Aug 2 at 16:23
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    I agree it's not totally random. I was trying to suggest an alternative which felt 'fair' in comparison to a perception of a dice being uneven in a game. If it's bad answer then It prob because I'm not sure what the problem is the questioner is trying to fix. Aug 2 at 21:01

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