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jk22

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Does this make sense in a frequentist approach since if i do the real experiment once and got 4 then the probability (statistics a posteriori) for 4 is 1 which is a dumb counterexample.

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- Thread starter jk22
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jk22

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Does this make sense in a frequentist approach since if i do the real experiment once and got 4 then the probability (statistics a posteriori) for 4 is 1 which is a dumb counterexample.

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billschnieder

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- #3

bhobba

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The frequentest approach the way its sometimes presented in beginning texts at the high school level is incorrect - its circular. What is probability - the ratio of a large number of trials. Why does it work - the law of large numbers - and around it goes.

The correct basis is the Kolmogorov axioms:

http://en.wikipedia.org/wiki/Probability_axioms

From that you can prove the law of large numbers.

When you apply it you need to make a few reasonableness assumptions such as a very small probability can for all practical purposes be taken as zero. You then apply the law of large numbers to events and say there is conceptually a very large number of trials where the outcomes are in proportion to the probability. The issue is the law of large numbers converges almost assuredly meaning for a very large, but finite number of trials, there is a small probability it will not be in proportion. But that's where the reasonableness assumption comes in - the very small probability of this being the case is taken as zero.

If you want to pursue it further the QM forum is not the place to do it - there is a specific sub-forum devoted to discussing probability. That said any good book on probability such as the classic by Feller will explain it:

https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20

Thanks

Bill

The correct basis is the Kolmogorov axioms:

http://en.wikipedia.org/wiki/Probability_axioms

From that you can prove the law of large numbers.

When you apply it you need to make a few reasonableness assumptions such as a very small probability can for all practical purposes be taken as zero. You then apply the law of large numbers to events and say there is conceptually a very large number of trials where the outcomes are in proportion to the probability. The issue is the law of large numbers converges almost assuredly meaning for a very large, but finite number of trials, there is a small probability it will not be in proportion. But that's where the reasonableness assumption comes in - the very small probability of this being the case is taken as zero.

If you want to pursue it further the QM forum is not the place to do it - there is a specific sub-forum devoted to discussing probability. That said any good book on probability such as the classic by Feller will explain it:

https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20

Thanks

Bill

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bhobba

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That's why these days the frequentest approach is based on the Kolmogorov axioms. The probability exists regardless of what trials you do. To experimentally determine it you need to conduct a large number of trials, so large that for all practical purposes it will give the correct probability ie you have decided on a very small probability you will take as being zero.

Thanks

Bill

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jk22

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- #6

bhobba

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Of course it would. The probabilities of QM are defined by the Kolmogerov axioms - not by circular frequent reasoning.

Thanks

Bill

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