Meaning and derivation of probability

[SudanBlack]

Hi,
I have recently been thinking about the fundamental meaning of the term probability, so I decide to discuss the topic with my tutor. He told me that the true definition of the probability of x occurring, P(x), is:

P(x) = Lim(Relative frequency of x in experiments) as n tends to infinity, where n = the sample size.

However, I have read many mathemetics textbooks which talk about "experimental probabilities" and "theoretical probabilities" - they refer to the definition I have previously mentioned as "experimental probabilities". "Theoretical probabilities" are apparently defined as follows:

P(x) = (Number of ways event can occur)/(Total number of events which can occur)

I wish to know if there is any thorough algebriac way to proove that the fraction calculated through the "theoretical probability" method is infact the number which the relative frequency will converge to as the sample size gets ever larger?

Also, I was curious as to what we would call the probability of any event for which the limit of the relative frequency does not converge?

Finally, is it possible to calculate the value which the limit of relative frequency will take, or can this only be obtained through repeat experiment?

Many thanks - eagerly awaiting replies.
Simon.

 

[Hurkyl]

However, I have read many mathemetics textbooks which talk about "experimental probabilities" and "theoretical probabilities" - they refer to the definition I have previously mentioned as "experimental probabilities". "Theoretical probabilities" are apparently defined as follows:

P(x) = (Number of ways event can occur)/(Total number of events which can occur)

That's not accurate -- this is only the probability if we assume a uniform distribution on the events.


There are two big issues here:

(1) Probability theory
(2) Statistical physics

Probability theory is a mathematical subject in its own right, and it takes a little bit of machinery to start doing things rigorously.

On the other hand, the use of statistics in physics requires some initial assumptions about things... and I expect that this issue is the heart of your question, so I'm going to move it over to the physics section to encourage them to take a crack at it.

 

[EnumaElish]

P(x) = (Number of ways event x can occur)/(Total number of EQUALLY LIKELY events which can occur)

I wish to know if there is any thorough algebriac way to proove that the fraction calculated through the "theoretical probability" method is infact the number which the relative frequency will converge to as the sample size gets ever larger?

My guess is there isn't because any algebraic proof necessarily belongs to the theory domain.

Also, I was curious as to what we would call the probability of any event for which the limit of the relative frequency does not converge?

Degenerate?

Finally, is it possible to calculate the value which the limit of relative frequency will take, or can this only be obtained through repeat experiment?

You can calculate it under certain (theoretical) assumptions, then perform repeated experiments to test whether these assumptions were justified.

 

[kosovtsov]

The probability theory is very nice logically consistent theory. The
hallmark of the theory is the notion of (statistical) independence.
It is very essential that for each problem in the framework of the
probability theory the initial (probability) measure is specified.

But in the physical statement of probabilistic problems we face some
fundamental difficulties (See, please, http://groups.google.com.ua/group/s...f4135cd854d0?hl=ru&ie=UTF-8&q=kosovtsov&pli=1).

 

[lotm]

Yeah, as people have said, it's worth disentangling two separate things:

1. The mathematical 'theory of probability'
2. The interpretation of said theory

The first (usually) refers to the consistent mathematical theory described by the Kolmogorov axioms. Other mathematical setups have been suggested (often in response to the philosophical issues - see below), but the Kolmogorov theory is the orthodox one.

The second is an extremely vexed philosophical question (see http://plato.stanford.edu/entries/probability-interpret/). Your tutor's claim that probability is the limiting frequency of an infinite sequence of measurements is one theory, but others abound, and there isn't really any widespread philosophical consensus. The idea of probability as the ratio of occurring events to possible events is the classical interpretation - it has the advantage of explaining the similarity of probability theory to other branches of measure theory (the theory of proportions, ratios etc); but as EnumaElish says, it requires that those events be equally probable, and you might wonder whether an account of equiprobability can be given that does not invoke probability (which would make the whole thing a circular explanation). For what it's worth, I'm inclined towards a kind of semi-classical interpretation of probability as ratios of 'volumes' in configuration space - the equiprobability of points in configuration space could perhaps be justified by noting the symmetries of such a space.

 

[linford86]

I agree with lotm. This is, philosophically speaking, a controversial subject area. However, the orthodox mathematical theory -- and, indeed, the one I know best -- is that described by the Kolmogorov axioms.