# Limited probabilities : a nonsense ?

Suppose i have an experiment which can give result 0,4 but that the probability p(4)<=1/sqrt(2).

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.

In the frequentist approach, the probability is the limit of the relative frequency as the number of trials approach infinity, so it doesn't make sense in the frequentist approach to apply probability to non-repeatable experiments or single iterations. This is why the frequentist approach is considered by many to be of very limited applicability.

bhobba
Mentor
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

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bhobba
Mentor
In the frequentist approach, the probability is the limit of the relative frequency as the number of trials approach infinity, so it doesn't make sense in the frequentist approach to apply probability to non-repeatable experiments or single iterations. This is why the frequentist approach is considered by many to be of very limited applicability.

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

This question was in fact linked to qm in the following sense : suppose you find the eigenvalues of bell chsh are 4 and 0 but that p(4)<=1/sqrt2 such that tsirelson bound is respected. Would this makes any sense ?

bhobba
Mentor
This question was in fact linked to qm in the following sense : suppose you find the eigenvalues of bell chsh are 4 and 0 but that p(4)<=1/sqrt2 such that tsirelson bound is respected. Would this makes any sense ?

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

Thanks
Bill