The meaning of probability and random variables

Question

I do understand pure mathematical concepts of probability space and random variables as a (measurable) functions.

The question is: what is the real-world meaning of probability and how can we apply the machinery of probability to the real situations?

Ex1: probability of heads for fair coin is 1/2. Which means that "if we will make exactly the same experiment over and over again, we will obtain heads roughly 1/2 of the time". More rigorously, there is some convergences under the hood etc. But this is nonsense: if I will make exactly the same initial conditions, velocity, humidity, wind etc, then I will always obtain the same result. Moreover, I cannot guarantee the same circumstances: we have some planets flying around, and molecules are not in the same places etc. So it is not really possible to make the same experiment twice. Like, you know, one cannot step twice in the same river.

Ex2: probability of rain is 0.9 for the next day. The "frequency" idea is really absurd in this case.

Ex3: I'm throwing a coin and see the side. You don't see it. I ask you, what is the probability that it comes Heads. This is really somewhat vague...

Answer 1

It's exactly the fact that the conditions can't be duplicated exactly that makes a probability model for the experiment viable.

With some practice, a person with good fine-motor control can become an expert coin-flipper who can fairly reliably produce either "heads" or "tails" as desired. The "fair coin" probability model might not work well for coin flips by such a person.

Answer 2

This is really more of a philosophy of math question than a math question. You might find this Wikipedia page interesting as a starting point: https://en.wikipedia.org/wiki/Probability_interpretations. I don't think there is one accepted "correct" interpretation of probability when it comes to the "real world."

Answer 3

We don't know well how to do the weather forecase, just because we don't know the exact independat variables nor their full relation between them. Our mathematical models are not accurate enough.

But we have zillions of daily measures: wind, temperature, moist, etc.
And we do this asertion: "For the N times we have observed the same measures as now, we've been right P·N times at saying 'tomorrow rains'". So, tomorrow rains with a probability of P.

This is just a small example of how stadistics were born as a useful tool.

Answer 4

General interpretation of probabilities is Bayesian: probabilities are subjective opinions or beliefs; of course probabilities must satisfy probability axioms, but this is the only requirement; any probabilistic model satisfying probability axioms is good. I believe it is helpful to think about probabilities as subjective opinions, but don't confuse subjectivity with arbitrariness. Subjectivity means that different persons may have different data and use different statistical models to infer probabilities.

How a probabilistic model corresponds to the real world is another question. Sometimes (flipping a coin) you can use frequencies to define probabilities, sometimes you need more advanced statistical models to infer probabilities corresponding to the real world.

Ex 3 demonstrates subjective nature of probabilities: you and your friend build different probabilistic models depending on information you and your friend have.

Answer 5

Wikipedia's definition of "randomness" is

the apparent lack of pattern or predictability in events

The word "apparent" there is important. Just because something can be predicted in theory, doesn't mean its not random. For example, a pseudo-random number generator in a computer program. It is without a doubt predictable, since the output results deterministicly from its seed. However in the general case where we assume ignorance of the internal state of the generator, each output in itself is considered random.

About the coin flip example, it is obvious in principle that given the right parameters and enough accuracy and precision, the outcome of a coin toss is predictable. This is where the question becomes a bit philosophical. What if we choose not to be aware of those parameters? Arguably the outcome is still random, at least to ourselves, but perhaps an independent observer could treat the same event as deterministic if they choose to make themselves aware of the information we ignored