I don’t have the time to go through all the analysis with this data, but there is an excellent Bayesian analysis of dice data in the scientific paper below, which quantifies not only how far from fair the die was, but also which of its physical properties were responsible - which is particularly impressive given that the die was rolled in the 19th century.
Mechanical Engineer Here. Models are used to predict that your test results will be good so you don’t waste time/money failing tests but realistic test results are the only true currency in the field.
The lack of squishy/meaty/living test subjects compared to your field probably gives us the luxury of meaningful repetition in tests.
Casino dice are cubes of a clear - usually red - material with white dots. The clearness means it is a lot harder to stick in a significant weight as you would have to match the colour and the refractive index to be invisible. The cubeness means you can use one die and any flat surface to check the cubeness of another die. Filling the holes fixes most of the difference in weight. You could fill it with something of a different density. It probably could not be a lot lighter, but you could could make it a lot heavier with something like tungsten, if you wanted to favour the ones.
Not very Game of Thrones, though. How about…
Get a giant ruby. Preverably steal the Red Eye from the statue of the Mad God, but Ebay is good too. See if they have any white star sapphires while you are at it. Cleave the ruby into into cubes. Polish them to roughly same height and width. Drill the holes. Fill them with the white star sapphire, which is a good density match. Polish the dice to the final parallel shape. Watch every dice roll while fingering your battleaxe.
Computer engineer here. I took statistics in college but remember jack and shit about it, and love all the detailed analysis you guys are doing. Boing Boing… where the really wonky smart geeks post!
I do stats professionally, and was sort of surprised the bias shows up with this few rolls–it suggests the bias is pretty strong. But note that if you look at whether one die roll predicts/influences the next, that also reaches significance p=.01 with the chi-squared test. This is akin to a comment Persi Diaconis made about finding coin flips were not independent when a math teacher did thousands with his class–the kids were not flipping coins very well. There is nothing about the dice that should induce non-stationarity/independence; I’d guess is is about the way it s picked up and dropped.
So, the problem with the chi-squared test on margins (what other people reported above0 is that the samples were not independent, which the test sort of assumes. The biggest outlier here is 6-6, which could skew the results of the margin test a bit. With relatively small N, it is shaky anyway. I wouldn’t be happy without at least 1000+ rolls per die.
Recommendation: Revise & resubmit–major changes needed. Revision should include second experiment that demonstrates bias with improved methods that don’t lead to trial-by-trial dependencies.
No you can’t and shouldn’t lump them together. (sorry @anon68287401)
The question asked was are these dice fair. The null hypothesis is that the ratio of the numbers thrown is 1/6 for each number or you could choose a different null hypothesis to prove that the proportions don’t differ on a single die.
Another question you could ask is are the dice the same. This is significantly different than asking if they are fair, they could each roll 1 all the time and be exactly the same, but you wouldn’t use them to generate your D&D character. The null hypothesis would be that the two die have the same proportions.
(This is a fun question I don;t have time to analyze because boring work calls to me.)
Hmm I wonder if one could roll them several times, compute them into a set of 3 numbers, and then enter them into the safe. You know, let Jesus take the wheel sort of thing.
Why not just use a shaker, and have the dice bounce off the back of a board after they come out of the shaker? That should take care of inconsistencies in how the dice are picked up and rolled.
It’s all well and good to calculate a p-value, but you can’t actually get P(A|B) from P(B|A) without more information. The odds of me winning the lottery if the sun comes up are pretty bad, but the odds the sun came up the morning I won the lottery (if such a morning were ever to exist) are spectacularly high.
No matter what math you do you’re always going to have to compare it back to your intuition about how reliable the process is. If I had similar rolls with a casino die, I would assume they were attributable to chance since I’m under the impression the defect rate in those is quite small. If there is a 2% chance your results were due to chance, you have to judge that against your confidence in hand forging.
The problem with even a Chi-Square test (at least, when it’s set up this way) is that it’s looking for bias, not looking for fairness. Failing to find bias is not the same as disproving bias.
A metaphor for the difference in the statistics would be, you’re looking for a needle in a haystack, and not finding it. What you need to do is prove that there is no needle in the haystack.
