Understanding probability, is probability defined?

[Stephen Tashi]

Or is it inevitable that we would use "belief" in explaining/justifying that we approximate a probability with a finite relative frequency?

If you have a mathematical theory that deals with one set of concepts ( e.g. weight, mass, position) and you try to apply it to a situation defined by a different set of concepts (e.g. price, value, utility) then you must introduce assumptions that establish some relation between the different concepts. You can introduce assumptions as formal mathematical axioms (which is necessary if you intend to prove your results) or you can introduce assumptions by your personal beliefs in an informal manner.

As far as I know, nobody has created a set of axioms for probability theory that establishes any deterministic relation between the relative frequency of an event and its probability. So if you want to establish such a relationship, you must do it using your personal beliefs - or else invent the mathematics that does the job.

 

[FactChecker]

There is nothing mystical about probabilities. Given a set of data, it is obvious, and dirt simple, to ask which fraction or percentage satisfies certain conditions. Scaling those numbers to a [0,1] scale or to a [0%, 100%] scale is just computationally convenient. Also, using the past events (statistics of past experiences) to anticipate similar future events (probabilities of future outcomes) is critical to the survival of even the simplest thinking animals.

 

[bobby2k]

If you have a mathematical theory that deals with one set of concepts ( e.g. weight, mass, position) and you try to apply it to a situation defined by a different set of concepts (e.g. price, value, utility) then you must introduce assumptions that establish some relation between the different concepts. You can introduce assumptions as formal mathematical axioms (which is necessary if you intend to prove your results) or you can introduce assumptions by your personal beliefs in an informal manner.

As far as I know, nobody has created a set of axioms for probability theory that establishes any deterministic relation between the relative frequency of an event and its probability. So if you want to establish such a relationship, you must do it using your personal beliefs - or else invent the mathematics that does the job.

Thank you, I now have the understanding I wanted about this subject. Sorry for taking so much time to understand it, but thank you very much for beeing patient!

 

[PeroK]

"On voit, par cet Essai, que la théorie des probabilités n'est, au fond, que le bon sens réduit au calcul; elle fait apprécier avec exactitude ce que les esprits justes sentent par une sorte d'instinct, sans qu'ils puissent souvent s'en rendre compte."

"One sees, from this Essay, that the theory of probabilities is basically just common sense reduced to calculus; it makes one appreciate with exactness that which accurate minds feel with a sort of instinct, often without being able to account for it."

Pierre-Simon Laplace, from the Introduction to Théorie Analytique des Probabilités.