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Roughly how many times do I need to roll a 6-sided die to feel confident that it's giving "fair" results? What about a 10-sided or 20-sided die?

Note that I will be actually manually rolling physical dice, this isn't just a textbook exercise. I'd like to minimize how long it takes me to perform this experiment with each die :)

I know this depends on my expected "confidence level" (95%? 99%?) If I choose a 95% confidence, for example, does that imply that 1 out of 20 fair dice will fail this test? Or that a single fair dice would fail the test 1 out of 20 times? If so, that sounds fairly high.

Are there standard techniques for doing this kind of a test?

Edit: It is beyond the scope of the math-focused question I've asked here, but I've explained more about the overall testing scenario over on the stats site here: https://stats.stackexchange.com/questions/14301/designing-a-test-for-a-psychic-who-says-he-can-influence-dice-rolls/14302#14302

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    Statistical tests checking if data came from a particular distribution should be used, e.g. [Andreson-Darling test](http://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test), [Kolmogorov-Smirnov test](http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test). If you are Mathematica user, [DistributionFitTest](http://reference.wolfram.com/mathematica/ref/DistributionFitTest.html) automates the testing.2011-08-15
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    You should also consider asking this question on [stats.SE](http://stats.stackexchange.com/).2011-08-15
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    I'm surprised to see someone mentioning Kolmogorov-Smirnov before mentioning the simplest sort of chi-square test.2011-08-15
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    Do you suspect a manufacturing/design defect?2011-08-15
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    The statistical test depends on the defect you are looking for. For example, two opposite faces more frequent could mean a slightly smaller dimension in that axis. Or less edge rounding on the face opposite a high frequency face. The statistical test will be more powerful if derived from a physical defect model.2016-05-07

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