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This is one of the problem taunting me over years while explaining probability.

In most of the high-school as well as graduate textbooks, there are at most very few lines to deal with this problem.

The problem is that the authors conveniently escape from explaining probabilities based on sample spaces (list of outcomes) to probability without talking about sample spaces at least once (deals with real-life events) after the definition of random variable comes in.

There will be no stress on such transition in most of the text books. Some students ask about the explicit sample space and events as sets. I say that they are no more necessary. But it became hard for me to teach the transition.

Some students randomly brings research papers containing random variables and asks for sample spaces and random experiments underlying.

Is there any universal rule of getting sample space during teaching the transition? Or making the students to understand that they are not necessary.

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    $\begingroup$ If you d on’t get a good answer here you might try stats.SE. My limited understanding is that probability has always had two interpretations, #successes/#outcomes (aka the frequentist point of view) and degree of belief (aka the Bayesian point of view). I cannot easily reconcile them either. $\endgroup$ – user2913 Sep 25 '20 at 14:47
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    $\begingroup$ I think the question would benefit from some contrasting examples of "probabilities based on sample spaces" and "probabilities without sample spaces". $\endgroup$ – Stephan Kubicki Sep 26 '20 at 0:08

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