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I've been using probability theory (, statistics, bayesian inference) for a while - and I find it very useful and mathematically elegant, but I still can't get where is the hidden "randomness" in a formal definition of random variable. I guess, there must be some "God with a Dice" that chooses which $\omega \in \Omega$ like "is going to realize". Or there's no such a thing and we always operate (and think of it as) just with numbers weighted by measure of corresponding (exists due to $\Omega$/$\mathcal B (\mathbb R)$ measurability of random variable) elements of $\Omega$ and no "dice inside"?

Or, in other words: how can something deterministic model something random? And if it can, where in formal definition that "randomness" is?

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Like most of mathematics, probability theory is a model of something real and physical, namely random processes. The mathematics itself is not random in any way, but it offers a way to understand real physical processes which are or appear to be random, or at least are conveniently modeled as such. So, when we perform an "actual experiment" like rolling a die, we realize some real, physical random process involving atoms and gravity and so forth. The way we model this process mathematically is through events (measurable sets) and probabilities (weights). In my opinion, the language of probability theory can be very confusing: random variables are not random, they model randomness. Random variables are just functions!

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A random variable is a kind of function that maps a given sample space to real numbers.

Which element or elements of the sample space is chosen is random.

For instance, if you have a list of names and corresponding dates of birth (a mapping from names to dates of birth) and you want to look up the date of birth given a name, then you have a deterministic (nonrandom function), but if you pick a name at random and then see which DOB comes up, then you have a random function. It is we humans who choose whether functions random or not.

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