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Regular dice are made such that opposite sides of the die add to 1+the number of sides. For example, a 20-sided die has 14 and 7 opposite of each other, adding to 21.

For certain types of games, "life counter" die are used. In these, sides are numbered sequentially, so that 1 is next to 2, which is next to 3, which is next to 4, and so on. You can see a picture here: http://www.coolstuffinc.com/main_supplies.php?fpid=Acc-QWSd20SpindownLifeCounterBluewhite

Now the question: generally speaking, will these two kinds of die yield equal probabilities? (Assuming that both dice are well balanced).

Particularly,

  • Will the probability of getting a certain number be the same?
  • Will the probability of getting a number above a threshold be the same? (i.e. rolling a 10 or more)
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migrated from stats.stackexchange.com Aug 10 '11 at 17:14

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Because this question concerns the physical properties of an icosahedral body, it is more deserving of consideration at physics.SE than on stats.SE. –  whuber Aug 10 '11 at 17:13
    
The answer must be related to how unpredictable (i.e. chaotic) the roll is. Anyone know much about how to specify such a thing? –  dmckee Aug 10 '11 at 17:25
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Alvaro: It is trivially true that if the side selection is truly random than re-numbering can not affect the probability. –  dmckee Aug 10 '11 at 17:33
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This should have stated at stats.SE for exactly the reason that @dmckee stated. We can answer it here because it's a trivial question, but the only "physics" part of this aspect is whether two identical polyhedra will behave identically. –  Colin K Aug 10 '11 at 18:04
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Dmckee's statement is basically a statement about definitions. The definition of a fair die is that all results are equally probable. The definition doesn't refer to the labels, so changing the labels has no effect on fairness. As an example where permuting the labels could make a difference, suppose you roll a die on a table, then bang on the table in order to disturb the die and get a fresh roll. Since the die may not be thoroughly scrambled, the second roll may be correlated with the first. But a correlation between two successive rolls is different from lack of fairness in a single roll. –  Ben Crowell Aug 10 '11 at 18:06

4 Answers 4

up vote 8 down vote accepted

If you want the high school answer, then no, the numbering does not matter. If all faces are equally-likely, the probability is the same regardless of how you number the die, and similarly all derived quantities (such as variance or probability to be greater than 8) are the same because the underlying distribution is the same.

If you want the real answer, it depends on the exact probability distribution of the faces of the die, which is best determined empirically (i.e. by rolling it many times).

If the die is weighted towards one face, so that face becomes more likely to be on bottom and the opposite face equally more likely to be on top, the the mean value of the normal die will not change, while the mean value of the life-counter die will change.

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5  
I don't really want to nitpick on a coherent answer to a vague and forgettable question, but why a Boltzmann distribution? With $\beta = 1$ in particular? It's not a thermal process is it? Wouldn't the more obvious distribution, though still flawed, be to weight each face by the fraction of the sphere it is the minimum for? This would give a much, much less sharp distribution than Boltzmann. If you don't have a good reason and just picked the Boltzmann dist. to demonstrate, it would probably be less misleading to state this, or just remove the section and say "its a hard problem". –  BebopButUnsteady Aug 10 '11 at 18:59
    
@Bebop Good point. I think that example was not very helpful, so I deleted it. –  Mark Eichenlaub Aug 12 '11 at 23:00

Mark Eichenlaub's answer is nice (although I am a bit dubious about the accuracy of the Boltzman-like formula). However, a little known fact is that even for a "fair" die, not all outcomes are equally likely. Research at Stanford and Santa Cruz (http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf) has shown that for coin flips, there is a slightly greater than 50% chance that a fair coin will come up with the same orientation as it started with!

With throwing a die similar physics is involved. Hence, with a "life counter" die, by choosing the die's orientation when thrown, one could more effectively bias towards higher or lower numbers than with a standard die.

The coin flip bias is only at around the 1% level (I have no idea what it would be for a 20-sided die), so whether this is significant or not depends on what level of effects you care about. But casinos make a living on advantages on the level of a few percent.

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As far as I know, there should probably be no significant difference. I can only imagine a small difference because the die would be slightly weighted on the side with the one. This is because, to make the die, numbers were cut out of the side. More was cut out for the 20 than for the 1. So if all the single digit numbers are on one half of the die and two digit numbers are on the other half, the two digit numbers might come up more often. I believe there should be no noticeable difference. I suggest you conduct an experiment to determine the truth.

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I think I can provide a different type of answer than the previous, if maybe in a less scholarly manner.

I am not an expert in the field of science, and certainly not in mathematics, however, I have rolled many an icasohedron in my day, be it d20 or hitdie. In my years of experience roaming the planes of imagination and roleplaying, I have found that, like it was mentionned earlier, a standard d20 will yield more "random" or "fair" or "what do you mean I rolled a 1 dang nabit" results (unless it's being rolled by that one guy who always gets a crit, but this falls within the realm of what I believe to be dark magic, so I will not speak further on that particular type of person and their dark gods). As also stated by others, I have found that a typical nonspecific cardgame hitpoint die will yield results closer to the previous value obtained, because that's how they roll.

Of course, I'm just speculating on the results I've seen over the years, and have no flashy diploma, peer review or even a spreadsheet of data to back this up.

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