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I find that sometimes in probability (geared towards non-math majors), the definition of the sample space will involve things such as:

"a set of possible outcomes of a random experiment".

I guess here "possible outcomes" and "random experiment" are not rigorous definitions.

A similar one can be found on Wikipedia: (https://en.wikipedia.org/wiki/Sample_space)

"the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment."

It just seems a bit odd to me that probability theory rests upon a common-place/colloquial understanding of what a "random experiment" is.

Can this "sample space" and in particular "random experiment/trial" definition be more precise (or is this a Principia Mathematica type of scenario, i.e. not worth defining it rigorously)?

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    $\begingroup$ No, the term "random experiment" is not precisely defined, and so in a perfectly rigorous treatment of probability that term is avoided (except perhaps in informal comments). Later, when you model some real world scenario using probability, you might introduce a sample space and you might think of each element in the sample space as corresponding to one possible outcome of a "random experiment" that takes place in the real world. But when developing probability rigorously you'll just say things like "A probability space is a triple $(\Omega, \mathcal F, P)$ such that so and so ..." $\endgroup$
    – littleO
    Jun 20, 2021 at 3:52
  • $\begingroup$ @littleO What is a random variable? At what point is randomness introduced when defining things rigorously? $\endgroup$
    – NicNic8
    Jun 20, 2021 at 3:57
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    $\begingroup$ @NicNic8 A random variable is a function $X$ that assigns a real number to each element of the sample space $\Omega$. Technically, we require that $X$ is "measurable", which is a term that's defined precisely in measure theory. The probability measure $P$ assigns a probability to each subset of $\Omega$ (technically, to each "measurable" subset of $\Omega$). So given a random variable $X$ we can ask questions like "What's the probability that $X$ is less than $a$"; in other words, we can ask what is $P(E)$ where $E = \{s \in \Omega \mid X(s) < a \}$. $\endgroup$
    – littleO
    Jun 20, 2021 at 4:13

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I think it is important to keep in mind that the sample space ultimately is an object defined by the user who is attempting to model a phenomena exhibiting uncertainty (or “randomness”). The sample space does not exist independently of this. Whether there is some true “sample space” for any random experiment opens a similar philosophical can of worms akin to problems like “does the number 1 exist.” Most would say that numbers themselves are purely conceptual constructs that we use to simplify/solve problems and communicate. The sample space is not different in my view.

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