|Jul28-12, 05:31 AM||#18|
How to verify dice by statistic means ?
You can't prove either way that the result is certain.
In hypothesis tesing you get two kinds of errors that relate to false positives (the result is false, but we have come to the conclusion that it's true) and false negatives (we come to the conclusion something is false but the reality is that it's positive) where positive means true and negative means false.
If we don't get a false positive or false negative then we have made the correct hypothesis.
There will always be the chance of a false positive and false negative and it's important to realize how significant these are in the context of statistical tests and hypothesis testing.
|Jul28-12, 09:20 AM||#19|
If you assume the die is unfair in a specific way (e.g. Pr(1) = 0.10, Pr(2) = 0.15, Pr(3) = 0.10, Pr(4) = 0.15, Pr(5) = .26, Pr(6)= .24 ) then you can test that hypothesis.
If you assume a specific "prior probability distribution" on 6 random variables that represent the probabilities of the outcomes of the die then you can compute the 'posterior probability distribution" for those random variables given the observed data. That would be the Bayesian approach. I think E.T. Jaynes actually worked this problem in one of his writings, perhaps it was in "Probability The Logic Of Science".
The use of the null hypothesis that "the die is fair" can be subjectively justified in many ways. "Innocent until proven guilty" is one way. "No reason to favor one face over others without considering data" is another. Sometimes the null hypothesis is touted as the "skeptical" hypothesis. Considering the imperfections in real dice, I would say the the hypothesis that "the die is fair" is a gullible hypothesis. However, most people are thinking in terms of "the die is nearly fair" without specifyiing quantitatively what they mean by that.
|Jul28-12, 01:38 PM||#20|
now I see a little more clearly.
But I came to the conclusion, that I will continue on this path only when forced to do so.
I just wanted to get a basic understanding of the concept (I guess I succeeded to a certain extent thanks to the posters here) but to go more into detail will be too time consuming for my situation.
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