[Math] Developing methods for determining the balance and reliability of dice.

[Psionicist]

In our group we have this peculiar green d20 that rolls 17 about 45% of all rolls. We also have an oddly shaped d6 with the annoying behavior of rolling 1 or 2 all the time. Superstitious people will call this lucky die, bad karma or whatever. I call it a fault in the manufacturing process, and I want to create a method how to find these faulty dice so only trustworthy and random ones are used in play.

A lucky die (such as our magic green d20) can transform the most worthless character, monster or NPC into something very mean, not to say devasting in hands of a DM when the encounter is already meant to be challenging for the poor PCs. As the current DM owns this particular d20, and I often use the bad d6, the problem is obvious. In a rules heavy campaign where you cannot, for example, stand up from prone and not provoke an AoO, I think it's odd that broken die are allowed whereas the most obscure actions that break the rules are not.

I am now mathematically trying to derive a fast and simple method and algorithm to find broken as well as balanced die, and I would like some suggestions how to achieve this.

One of the most obvious methods is of course rolling a particular die as many times as possible to check if each side is responsible for an equivalent percentage of the total outcome (say a d20 where 4-6% are 1:s, 4-6% are 2:s, 4-6% are 3:s and so on…). The questions are then, how many times one should roll, how to decide if a die is good or bad based on the figures, finding the right fault tolerance (as no die is 100% perfect today or for the matter that no one wants to roll more than say 50 times to determine the characteristics of the die) and so on and so forth.

All ideas and comments appreciated.

 

[GnomeWorks]

wrong link... hold on...

 

[GnomeWorks]

Better link.

 

[physics_ninja]

A real quick and dirty way to test this is to roll the die about 10 times for each side that it has. Roll the d20 about 200 times and the d6 about 60 times. Each number should come up 10 times but can vary from 5 to 15.

If you get results exceeding that range you'll want to do a chi-square test. As it turns out there is an article "Be Thy Die Ill Wrought?" in Dragon Magazine. I don't know the issue number but it is the October, 1983 issue.

Keep in mind though, if you really are unlucky, then the die will behave properly every time you test it, but not when you are actually playing.

Those dice must be punished. Quickly and brutally.

 

[Norfleet]

Dice are inherently nonrandom. The process which governs a die roll is not actually a random quantum process at all, but a physical process that most people lack the ability to effectively control. Controlling a d4, for instance, is only a DC13 dexterity check, making it a trivial feat for anyone with a dexterity greater than 14. Controlling a d6 is slightly trickier, requiring a DC17 dexterity check. Controlling a d20 is quite difficult, though, as attempting to force a d20 to get the result you want requires a DC31 dexterity check.

 

[abri]

So what you have in theory is a random event with 20 equiprobable possibilities.
You are trying to see if a random event obey this distribution.
You need a significant population of results for each possibilities of the test.
So on a linear distribution (theoritical d20) this mean you need 20 event in each possibilities to have a 97% probability of the test being the correct distribution:
so basically you need to roll untill you have a minimum of 20 of each results (so in theory 400 rolls).
But really that's overkill, since it will give you also an aproximation of the random distribution of the die...

 

[nopantsyet]

You look those dice straight in the eyes...you can spot a bad one easily enough. Those?....well....they must be destroyed ruthlessly as an example for other would-be malcontent dice. They'll learn quick enough. And if they don't........

You could always do a high-resolution sonic analysis of them, like they do to test bridge concrete. That would reveal any cracks or areas of nonuniform density. From there it's simple physics to determine where it will fall, at what angle it will strike, and where it will ultimately stand, given a certain velocity and trajectory.

Generally if you can't spot any trends, it's random enough. Part of the perception of randomness is not just the result itself, but the ability to predict the result. And it's the perception of randomness, not the actuality of it that provides the thrill of the hunt.

Edited: Spelling...grammar...what an idiot?

 

[tarchon]

I am now mathematically trying to derive a fast and simple method and algorithm to find broken as well as balanced die, and I would like some suggestions how to achieve this.

Hypothesis testing, statistical process control. See any statistics book.

 

[Eldragon]

The mathmatical methods are time consuming. The best way to test a die is a very simple phyical test.

Use a device called a "Dice Balancing Caliper". Casino supply stores sell them. You clamp the offending die between the calipers and you give the die a spin. If the die comes to rest smoothly then you know it is not weighted, if it wobbles as it slows down you know its loaded. It wont work on a d4, and on d12 and d20 you are going to need to clam and spin it multiple times (because there are so many different faces on them).

Most cheap dice at a FLGS are weighted in some way, aka "Loaded".

I like to determine which dice of mine are weighted, what number they tend to roll, and use them on my PCs. (I make all my rolls public as a DM)

 

[Umbran]

IIRC, in an issue of Dragon long, long ago, they included the code (in BASIC, I think) for a program to perform a Chi-squared analysis of dice rolls. Someone with the Dragon CD-ROM might look it up for you.

In general, there are few fast ways to perform the test you want. Physical measures need specialized expensive equipment (those calipers are not cheap, I'm told). Mathamatical tests require a largish number of rolls of the die in question.