My boyfriend and I have agreed to leave this question up to the Dopers as we cannot agree and must salvage our relationship somehow.
He recently acquired a die, a d20 to be specific (a die with 20 sides). He claims that the die can’t be used for rolling purposes because it is not configured in the traditional way (ie, 1 being opposite 20, 2 being opposite 19 etc.) Instead, it has the high numbers on one side and the low ones on the other.
I claim that (unless the die is improperly weighted) it will still be random, as each side should still have a 1 in 20 chance on being landed upon.
He claims that while the number itself would still be random, you’d be more likely to get a higher or lower average. Why else would they make all the other dice that way? he says.
I’d say that although there should theoretically be a random chance, that it would be much easier to intentionally roll a high number if its arranged like that. Seems like a needlessly poor design to me.
Your boyfriend is putting forth a contradiction. It is impossible for both the number to be decently random and the average to be higher, as long as each number is represented only once. The die, assuming it is properly weighted, geometrically sound, and honestly rolled, will be just as random with any arrangement of numbers.
On the other hand, with some arrangements of numbers, it’s rather easy to deliberately “roll high” or “roll low”, as desired. On my d12 (12-sided die), for instance, the 7, 8, 9, 10, and 11 are arranged around the edges of the 12, with complimentary numbers on the opposite faces. This means that I just have to aim for the 12, so to speak, and I’m likely to get a number in the top half. The easiest way to prevent this sort of abuse is actually to make the die such that complimentary numbers aren’t on opposite faces of the die. If, for instance, 20 is right next to 1 on your boyfriend’s die, he can’t “aim” for 20, for fear of accidentally getting a 1.
You can AIM for a specific number/side of the die? This is even possible? Has the ability to do this been proven somewhere using an actual scientific methodology? I am skeptical.
it is easier to intentionally role high or low but if you roll it randomly it will be like any other d20, random. it’s only if you try that the randomness will go.
I’m talking ridiculously small differences here, but this might be the source of his concern. On high-quality dice, I imagine that the weight of the indentations representing the numbers bored out is equal to the weight of the extra paint applied there. If not, however, either the high, 2-digit numbers, or the low, 1-digit numbers would have heavier faces. If the lighter faces were clustered on one side, it might make that side more likely to land up.
If the high numbers are really all on one side it might be possible to practice rolling so that you’d come up with higher numbers most of the time.
This brings up the question, what are you using the 20 sided dice for? In the nost popular use for such dice, Dungeons and Dragons (and other role playing games) the dungeon master can make a roll for the player. Ususally, this is done in case that the DM doesn’t want the player to know if they suceeded or not. But the DM can do all the rolls, although he should tell the player to stop or make the player take other actions (such as rolling from a cup) that would negate the effect.
That’s just like a 6-sided die: the high numbers are all centered around one corner, and the low numbers are centered around the opposite corner. So the highs are on one side and the lows on the other.
The 1-6, 2-5, 3-4 opposites arrangement on a d6 is purely stylistic. So a 1-20, 2-19, etc. one for a d20 would be too.
If you get a higher average, then it is not a random die. If one range of numbers is rolled more frequently, it’s a design problem.
When he and you has a free evening, roll the die a few thousand times, recording all the results (on a computer preferably). The average should be 10.5. Forgetting my statistics, but there is a formula for determining how off the average can be for a certain number of rolls and still be expectantly random. I.e., an average after 3 rolls of 15 is not suspicious (14, 18, 13 for example); but for a few thousand it is.
P.S. - Matter of fact, I just did a simple spreadsheet with 4000 random rolls of a 20-sided die. The average never went out of the range 10.0-11.0. So, without the math formulas to use, I’d say roll it 4000 times (not too hard for a single evening). If the average is less than 10.0 or greater than 11.0, I’d say somethings is indeed fishy with the die.
AWB,
So you did one replication study of 4000 rolls (N=4000)? How many replications did you do? Monte Carlo studies of less than 10,000 replications are often inaccurate (p>.05).
Of course, you are correct in the average computations. And, of course, one doesn’t need to do 10,000 replications (You’d need more than one evening!) to show that the rolling is random.
As stated above (Trucido) the chance of rolling a specific number on a 20 sided die is 1/20. The assumption made by BF is that a roll is more likely to be high if the die is on the high side. While this is quite true, one must consider that the chance of the die being on the high side is 1/2. Both of these events are random.
As far as “aiming” the die, I guess it’s possible, but a d20 is darn near spherical. I doubt one could control it as easily as, say, a d12 or d8.
Yes, d6 have high and low numbers “grouped” but I’m not sure if that argument holds much weight here given that the center of these numbers is a corner and not a face. I’ll have to think about this some more.
Speaking as a GM who has gamed with some shady characters (bwah hah hah), I’d be suspicious of a d20 with all the high numbers on one side. All you have to do is file down the edges that the high numbers share with an emery board just a little tiny bit and suddenly you’ve got a high-roller.
There are only two possible sources for your friend’s worries, and both are poorly founded.
(1) the actual weight distribution may be off because of the indentations that form the numbers and/or the paint. The difference here is incredibly negligble.
(2) Robustness. If, for whatever reason (poor manufacturing standards?), the die was slightly weighted towards one half or another, by grouping like numbers together, you would increase the likelihood of those numbers. If the numbers were set up to mix high and lows, this wouldn’t be a real factor, since weighting it towards one side would be as likely to result in a high number or low number. However, this would be very striaghtforward to check, just roll it a bunch of times and see what happens.
BTW, some translation here… “He claims that while the number itself would still be random, you’d be more likely to get a higher or lower average.” Since people have seemed to pick on an alleged contradiction here, I’m going to defend him.
I belive what he is trying to say is that the die may be (for example) weighted towards the “low numbers” half. In this case, you would be more likely to get a lower number. You would still not know exactly what number was going to come up in the next roll. This is probably what he (or you) is referring to as random, and shouldn’t be consued with the concept that the next roll has equal probabilities for all outcomes. Random outcome does not neccessarily mean equal chance of all outcomes. Flipping two coins has a random outcome, but you’d be more likely to average exactly one head than two heads (2:1).
