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Today our professor did some examples of sample spaces for different problems with dices and said for each example in way like he had just proved something: "This is the correct sample space for the problem".

What bothers me is the following: How can someone say that this or that sample space is correct one like he would be proving something ? AFAIK one can't prove or disprove that this or that sample space is "correct" since the experiment takes place in reality and there is no way to prove anything about reality (merely by the act of modeling we make reality accessible to proof). The choice of the sample space always remains arbitrary; one can only give plausible arguments (for example by doing a large number of experiments and seeing if the "empirical probability" of some event coincides with the "mathematical probability" of event in a given sample space. If it does, then the choice of the sample space was "correct"), why one should choose one sample space over the other.

Would you consider my reasoning to be acceptable ?

(That question maybe has something to do with this one).

P.S. Please keep in mind, that I this course is about elementary probability.

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The assignment of probabilities has only an indirect connection to the sample space: probabilities do not come attached. Thus a large part of your argument does not apply. However, I would agree that there is no unique correct sample space. –  André Nicolas Mar 6 '12 at 17:54

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In any statistical problem you should be aware of any assumptions you are making. So in an elementary probability problem about tossing two dice, you may assume that the number you get on the two dice is independent and that each number 1,...,6 is equally likely.

If these assumptions correspond to the real world of tossing two dice is largely irrelevant. You could test the assumptions for two particular dice using a standard hypothesis test. But you could never 'prove' in the mathematical sense that your dice are independent and fair.

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