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Kolmogorov's axioms specify a set of elementary events $E$ which is the sample space in modern theory. He then specifies a field of probability which is the power set of $E$. The axioms state $P(E) = 1$ and to construct a field of probability all the elementary events of $E$ are assigned probabilities.

My question is why does some of the modern literature and problems exclude specification of the elementary events and probability for each elementary event? There is also no specification of the sample space.

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For instance, I'm reading the following problem and keep asking myself: "What is the sample space?" "What are the elementary events and their probabilities?". Is this information not required? Thanks

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This doesn't say that you must specify these events, only that this is an example of "the simplest fields in probability." You can define them in this way if your universe only contains a small number of random variables with finite support, but in general it is not a very useful construction (it won't help you handle normal random variables, for example).

In general, it is very rare that you have to explicitly construct the probability space. For almost any modeling or computational question, and most theoretical questions, you usually just assume that there exists a probability space $\left(\Omega,\mathcal{F},P\right)$ with abstract elements $\omega\in\Omega$, but in the analysis you work with events and distributions related to random variables. The only time you would ever have to really assign some specific meaning to $\omega$ is if you were designing some kind of abstruse counterexample.

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