# Dice Rolling Method Fairness

Recently, in D&D sessions I have been rolling poorly (nothing over 10), and thus tried ways to improve my rolling. I was using a plastic solo cup last session, and got really good results compared to normal. I would place the die with the 20 facing down while holding the cup upright. I then spin the cup in a circular motion, slowly tilting it to about horizontal, then slamming it down on the table to get the result(kind of like Yahtzee). I don't personally think this has any real advantage to rolling normally, but one of my friends said that rolling in a consistent method would essentially make the rolls less random, which would be considered cheating. I know some people will not let others touch their die, or start all their rolls with a 20 facing up, but I thought that was more superstition than a tangible change. Recently I found this article that rolls would not be completly random but it doesn't notate any specific methods used for the motion of the dice. It does imply that changing the situation in which the dice are rolled can affect the outcome or probability.

So my questions are:

• Does the physical rolling method change the odds of the dice in a significant way?
• Is simply seeking for a method to improve rolling considered cheating?
• Hi folks, please reserve comments for requesting clarification or suggesting improvement. The comments posted on this question or its answers have gotten very discussion-y and uncomfortably far into answers in comments and have been removed. Further chatty comments will be removed without warning. – doppelgreener Nov 13 '17 at 15:44

## This is a classic clustering fallacy

Wikipedia defines the clustering fallacy as:

...the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random.

Basically, a perfectly random die will have long (potentially very long) steaks of seemingly non-random behavior, either very high or very low. Therefore, your observation of low rolls in multiple sessions could simply be true random variation. Streaks, both high and low, might not be common, but you'll definitely notice them a lot more than more varied behavior. After all, a sequence of exactly 19,20,19 will be a lot more noticeable than a 3, 16, 9, even though the exact probability of both sequences is the same.

The only way to truly know if any method or die is biased or not is to do dozens of identical rolls and look at the resulting distribution. Additionally, everything from the die itself to the rolling surface to your throwing technique is going to affect the roll.

## How random does it need to be?

Moreover, how random is random enough? If you read the paper that your article is citing, they say that for a 6-sided die that bounces 4-5 times, the probability that the die lands on the face that's the lowest at the beginning is about 20%, whereas the expected random value is about 16%. Would you notice a 4 percentage point difference on a d6 roll?

For a more concrete personal example, I have a gold d20 that that has a very strong bias toward rolling 20s, according to the saltwater flotation method. However, when I use this "golden" d20, I don't actually notice a greater proportion of 20s showing up in my rolls.

Therefore, the answer to your first question is yes: rolling methods do affect the roll, but unless you're using actual loaded dice, the difference is too small to matter. This means that the answer to your second question is also no, as well.

• I would like to add that one rarely notes "all rolls" - one will tend to remember more important rolls more clearly. Which is hard to verify and counteract. I suspect if someone noted all your rolls during the session and did some maths, you would find you haven't rolled that badly during your "bad streak" and not that amazing after you changed your method. – bytepusher Nov 13 '17 at 22:05
• You don't need thousands of rolls. Illmari Karonen posted a pretty good run through on the chi-squared method here a while back. I've linked it, and I recommend y'all read it and upvote, if you haven't already ;) In any case, your general point in that sentence was right, it's just that you need dozens, not thousands. – Please stop being evil Nov 14 '17 at 6:46
• I would also like to point out that some cheaper dice sets (a buck or two for the set) are often made of cheaper materials or aren't checked as thoroughly for balance resulting in unintentional "loading", whether it be in your favour or not. Things like bubbles in plastic curing processes would cause this. Please see here for more information. – Nathan Smith Nov 14 '17 at 22:24
• @NathanSmith That's the point of my second to last paragraph, though: while the dice might technically be unfair, it often makes no difference in actual play unless they're actually deliberately loaded. – Icyfire Nov 15 '17 at 1:26
• "Streaks" are common - particularly if you just think of your dice as high or low. As a rough guide, when flipping a coin n times you would expect a streak of sqrt n - so 10 heads in a row in 100 tosses. – Dale M Nov 15 '17 at 8:59

You've got a fancy dice-rolling ritual, and your friend is concerned it's cheating, and you're wondering whether it really impacts your roll at all.

We consider dice rolls to be "fair" or "not cheating" when there was a roughly even chance of them landing on any given face. So, I'd like to ask you a different question: if this ritual didn't improve your dice rolls, why are you trying to do it?

There's really only two possibilities:

• This ritual has no practical impact on the outcome of your roll, and thus it's just a pointless time-wasting ritual. (One that stresses out your friends because it might be doing something.)
• This ritual does have practical impact on the outcome of your rolls, and either you're artificially making your rolls better (which is cheating), or you're artificially making them worse (which sucks for you).

At this point, whether it does anything for you is academic: it's better off just not doing it, because both scenarios are undesirable.

# Let's take a look at what we mean by cheating.

Generally we'd consider it cheating if I used rigged/weighted dice that always came up 6:

We'd also consider it cheating if I put them down on the table, pushed them over with my fingers until the 6's faced up, then called that my roll.

This is because I took care to rig the outcome to exactly what I wanted, rather than actually follow the spirit of getting a random spread of results.

We'd likewise call it cheating if I put the 6's face-down in a cup, then swung the cup over with a practiced technique that'd make them all fall out with the 6's facing up. Your method of putting the d20 down and spinning it before slamming the cup down sounds a lot like that technique.

This means... rituals to improve your chances of good rolls (to above average) are cheating. If they weren't improving your rolls, there's no point doing them, and you're just stressing people out.

# Look for a way to give yourself even results, not improved results.

Usually rolling by hand is sufficient: let it roll out of your hand and across the table such that it actually rolls. If you don't have much space to work with, using a dice tray can be helpful.

If you have a dice cup, you'll get even rolls by shaking the cup around a few times and then turning it upside down for the result.

If you're concerned about getting continuously poor rolls, instead take the dice out of your hands, literally: use a dice tower. These are specialised devices which force the dice to roll through themselves chaotically before the dice come out at the bottom to give you your results. You simply drop them into the top and let the tower do its work. These come in all kinds of shapes and sizes.

In short: Just focus on rolling fairly. Don't use tricks to improve your rolls, you just stress your group out and you're cheating if it has any practical result.

The method you describe could be used, with skill and practice, to increase the incidence of rolls that come up 20, or numbers adjacent to 20 on the faces of the die.

To get convincingly random rolls, drop the die into the cup, allowing it to bounce several times, and not looking in at it or turning it to any specific position. Put your hand over the mouth of the cup, shake vigorously, and roll the die out onto the table.

Dice rolling?

## It's Math & Physics Time!

Ok, not in the sense of digging out a few formulae and then calculating a lot, more in the way of analyzing the problem:

Dice are well defined polyhedral bodies. They can rest on one face only, making the results well defined. But what does happen before they roll and stop?

We give them some kinetic energy. Here physics comes in: as long as all parameters are exactly the same, physics dictates that the result (well defined number) should be the same. Now, we need to look what parameters are affecting the roll. And in this case, it is actually more than the eye can see.

• Starting parameters of the polehedron in question:
• Face it sits on (F)
• Position in cup (X, Y)
• Rotation of the face in relation to the inner coordinated of the cup ($\Phi$)
• Starting parameters of the cup:
• the material of the cup
• the shape of the interior surface of the cup (C)
• rotation of the cup in relation to the outer coordinates ($\Psi$)
• angle of the cup floor before starting the spinning ($\Theta$)
• movement formula of the spin-up process
• This is most likely a very very long set of terms, as your muscles work uneaven and non-repetitive
• It is also the most influental part of the tumbling, as the sheer ammount of entropy introduced in this (faster/slower on any step of the path) far outweighs the entropy of the starting parameters.
• movement formula of the cup urning over
• see above
• movement formula of the die finally rolling over the table and impacting the walls
• see above

So... we got 1 set with a well defined starting parameter (F) and two somewhat less well defined starting parameters (X, Y $\Phi$), in a not perfectly well defined cup (Surface shape C, material) without checking for the starting parameters ($\Psi, \Theta$) of it.

It is possible to predict how the polyhedron impacts the first wall to a good degree... then it does impact the second one using the modified parameters... then the third...

With each impact inside the cup, the ammount of entropy (unceartenty) about the dice's properties grows. We can easily say "the dice is in the cup" because that is the outer constraint of the experiment. We can even say "it is very likely in this area of the cup" because that is where it impacts the wall the most times as we feel it in tumbing. But trying to make an educated gess of the highly entropic tumbling dice inside the cup is something that makes computers boil. The process simply increases its entropy with each impact to some degree, and by using imperfect mmanipulators to introduce kinetic energy within a spectrum depending on the movement (hands!) we can't even pinpoint some very cruicial parts of the movenmet formulae of the cup and the dice...

Let's just say that once the polyhedron in the cup interacts with the walls a few times, the entropy that induces becomes large enough to make it hard to distinguish to a chaotic system.

But if you can snap a photo of the moment the die leaves the cup to tumble onto the table... at THAT moment it is somewhat possible to predict its movement and finally landing position.

## tl;dr:

The entropy introduced by the spinning/rolling motion far outweighs the reduced entropy of setting one parameter in the experiment, when all the other parameters are not well defined.1 So no, unless you are NS-2 and are able to repeat the exact positioning and movements all the time, you do not repeat an experiment like a physicist or mathematican would. When they conduct such an experiment, they try to limit the order of freedom for each parameter they know to a minimum.

Or, to quote an article about rolling dice from some kind of apperature:

Practically, the predictability can be realized only when the die is thrown by a special device which allows to set very precisely the initial conditions. [...] If an experienced player can reproduce the initial conditions with small finite uncertainty, there is a good chance that the desired final state will be obtained.

You are not NS-2, you don't use a special rolling device to set up ypur rolls and perform them, so no, you are not able to get to this level of small finite uncertainty to make reliable predictions.

1 - "In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value." - a well defined function would be $f(x)=x+1$ or even $f(x)=\text{constant}$. Do you want to know more?

• If you're going for a full on scientific explanation you should really mention chaos - ie that a small change to initial conditions can result in a large change to the final outcome. This is particularly important because in an experiment like throwing a ball small changes to things like how hard you throw due to imperfect reproduction of the test result in very limited changes to where the ball lands. You still get all your balls landing in a small area. In a chaotic situation (as I believe rolling a dice is) the small changes very quickly result in an impossible to predict outcome. – Chris Nov 13 '17 at 12:33
• @Chris Not well defined parameters are also called "order of Freedom". The distinguishing between a "very entropic" and "chaotic" system are hard to find though. – Trish Nov 13 '17 at 15:52
• I've not come across the use of entropy in this context before so just assumed it meant the amount of variation from the planned initial conditions. I have to admit that your comment has just confused me slightly since "order of freedom" doesn't seem to be usefully defined by a google search or did you mean degree of freedom (in which case I'll agree that any parameter of the system is a degree of freedom but am not sure what you mean by mentioning that)... Basically I think I am stuck on not really understanding the sense in which you are talking about entropy in the system... – Chris Nov 13 '17 at 16:03
• @Chris Eh, degree of freedom, yes... Any non defined parameter (degree of freedom) can be assumed to have any legal value for the calculation. This means that the uncertainty (entroy) of the first parameterset is not small. – Trish Nov 13 '17 at 16:19
• It is the use of entropy in statistical mechanics btw. – Trish Nov 13 '17 at 16:39

Is it cheating? Whether or not it meets the technical definition of cheating, I would still say yes.

Let's just assume you actually are affecting the roll of the dice. Why do it? Isn't part of the fun of the game to risk missing a hit, or even dying. If you can affect the outcome to your advantage, why bother playing? Its no different than turning on "god mode" in a computer game.

• Is it cheating if it's just done to overcome the way that the dice are cheating already (by rolling low all the time)? Tongue in cheek. – Beanluc Nov 14 '17 at 0:26