Originally published at: http://boingboing.net/2017/01/17/irolledthesehandforgedmet.html
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I rolled these handforged metal dice hundreds of times. How fair are they?
Well those are medieval dice. Maths was all different back then…
All else being equal, the divots aren’t filled which automatically gives the 6 a slight advantage.
Get some calipers and a good square and check the actual measurements. They may not be cubes.
Float them in a glass of mercury and see how they settle? @AcerPlatanoides has pics!
Also need to check the consistency of the edges. Having some edges rounder than others affects the outcome, too. One of the reasons that float tests are not dispositive.
The null hypothesis is that the outcomes you experienced were as a result of chance, rather than due to biased dice. Statistics can tell you the chance that the results are actually random is true. A low
probability that your results are random implies biased dice. For example, all sixes is vanishingly unlikely to be due to random outcomes. But what is “low”? Depends on the consequences of you being wrong.
Hmmm…
Specific gravity of Mercury: 13.6
Specific gravity of Steel: aprox 7.8
Huh, that actually works…
Might be easier to check if we had a bar chart showing the number of times each digit was rolled.
Here you go. Sort of.
111111111111111111111111111111111111111111111111111
222222222222222222222222222222222222222
33333333333333333333333333333333333333
44444444444444444444444444444444444444444444444444444444
5555555555555555555555555555555555555555555555555555555555
6666666666666666666666666666666666666666666666666666666666
111111111111111111111111111111111111111111111111
222222222222222222222222222222222222222
333333333333333333333333333333333333333333333333
4444444444444444444444444444444444444444444444
555555555555555555555555555555555555555555555555555555
66666666666666666666666666666666666666666666666666666666666666666
Doesn’t look like they like the middle numbers.
Cheers! I’ll edit my post to include this
As N increases, so does the margin of error. 1500 would yield an margin of error of ~2.6%. 10k would get it down to 1% and that would be an excessive amount of work.
You also might like to look at the mode. That would show you what number comes of the most often, as dice rolls are generally counted as integers. Hypothetically one could have dice that show as being fair, when reality they are rather biased. Looking for the difference between the number of times each of the sides comes up would show this. It shouldn’t come up to zero difference between each possibility, but it should not be great also.
A quick way to find the mode is to use a concordance tool or charcter frequency tool, (Or just use the stats tool in Excel or what ever program you are using). In the link I provided, select “all characters” and you won’t have to do much data cleanup or formating.
I don’t expect the dice to be fair. I’d like them to come up 6 more often but that’s me and the GMs I used to play with were big on roll playing and strategy and loose on requiring fair dice. Nobody like a cheater, but nobody complained when the dice rolled more favorably. (Punishment for egregious and calculated jackassery by one DM was done by a d100 that was terribly biased to lower numbers.)
This isn’t the right statistical test for randomness. You need chisquared.
Are any of those outcomes safe combinations?
Here they are in their opposing side pairs:
A
111111111111111111111111111111111111111111111111111
6666666666666666666666666666666666666666666666666666666666

222222222222222222222222222222222222222
5555555555555555555555555555555555555555555555555555555555

33333333333333333333333333333333333333
44444444444444444444444444444444444444444444444444444444

B
111111111111111111111111111111111111111111111111
66666666666666666666666666666666666666666666666666666666666666666

222222222222222222222222222222222222222
555555555555555555555555555555555555555555555555555555

333333333333333333333333333333333333333333333333
4444444444444444444444444444444444444444444444

A DM who is also a math professor blogged a testing method a while back that uses the chisquared method and is relatively easy to do manually.
Well, if random spreadsheets off the internet can be trusted…
We never did this test when I took statistics
I ran the numbers using the “chisq.test” function in R. The pvalues are 0.126 for Die A and 0.172 for Die B. Neither of these is nearly significant enough to support a conclusion that either die is biased. For reference, a pvalue of 0.125 means that you would get a result at least that suspicious from a fair die in 1 out of 8 trials.