# Talk:Probability axioms

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## Earliest discussion

Would it be possibvle to rejig this in terms of Borel sets? Just a more elegant way of expressing it IMHO; here it's as if the defintion of sigma algebra comes out of probability theory. It might be moe confusing for the general reader however.

It's more confusing for me, anyway. The universe S is not necessarily a topological space, so what is a Borel set in this context? Fool 23:42 Mar 11, 2003 (UTC)

"Fool" is right. Borel sets are by definition members of the sigma-algebra generated by a topology. But there need not be any topology on a probability space. Or, at least, no topology is explicitly contemplated by the conventional Kolmogorovian definition.

I have moved this article to "Probability axioms" (plural!). Usually it is better to use the singular than the plural in the title of an article; "zebra" is better than "zebras". But in the case of this article, it is colossally silly. This is not about axioms as individual things; it is about systems of axioms, or, at least, about one particular system of axioms --- the one formulated by Kolmogorov. Michael Hardy 00:41 Mar 12, 2003 (UTC)

I think Cox's axioms should be stated here in addition to Kolmogorov's; if I'm not mistaken, the Kolmogorov axioms are derivable as theorems from Cox's -- which, again IINM, was Cox's point, that the accepted laws of probability are derivable from more basic assumptions. The one bit that doesn't carry over, if we start from Cox's axioms, is countable additivity -- IINM finite additivity derives from Cox's axioms but not countable additivity. Wile E. Heresiarch 15:30, 27 Dec 2003 (UTC)

I believe Axiom 1 should be stated simply as P(E) >= 0 rather than 0 <= P(E) <=1, since P(E) <=1 is actually a consequence of P(S) = 1, P(E) >= 0 and countable additivity.

## Sigma-algebra

I'd like to rework this article to at least begin in a less technical manner, as I distinctly remember being confused by this when I learned it. But I'm not sure about the history here. I can understand why a modern mathematician working in ZFC needs this sort of thing to excommunicate non-measurable sets, but did Kolmogorov really care (or even know) about such things at all? Did his first axiom not apply to all events? Was his third axiom really countably additive and not just pairwise (therefore finitely) additive? -Dan (Fool) 03:47, 5 December 2005 (UTC)

Yes, his celebrated 1933 book does explicitly state the axiom as countable additivity. And he said that that, rather than just finite additivity, was merely for the sake of convenience. That Kolmogorov did NOT know about such things as sigma-algebras and non-measurable sets strikes me as implausible, but I haven't looked that closely. But after all, Kolmogorov's book appeared in 1933, so one should expect it to be quite modern in approach. Michael Hardy 22:11, 5 December 2005 (UTC)
Thanks. Actually, I figured it out, I was confusing him with Kronecker for some reason. Oops. -Dan 00:44, 6 December 2005 (UTC)

## Proposed merge with probability theory

I propose this page has a merge with probability theory. Please add your comments to the proposal there. Thanks Andeggs 16:13, 24 December 2006 (UTC)

If someone can check that all of the relevant information has been moved to probability theory, we might be able to delete this page and replace it with a redirect? MisterSheik 17:12, 28 February 2007 (UTC)

I'm not for the merge anymore given that this page is linked to from probability space. And I'm not for folding probability space into probability theory given that the treatment at probability theory has a nice parallel structure with the other kinds of prob. theory, while the one at prob. space is more in depth, and clearer given no prior knowledge. Yeah, there's duplication, but there's also a lot of duplication of probability distribution, for example (which could really use a clean-up.) MisterSheik 17:41, 29 March 2007 (UTC)

## Oversimplified?

From the discussion here it looks like this once discussed sigma-algebras and now doesn't. As it stands, this is just wrong - not every subset of omega can be assigned a probability. — ciphergoth 10:29, 11 November 2007 (UTC)

## Question regarding third axiom

I have a question regarding the third axiom. P(E1UE2......UEN)=P(E1)+....P(EN) thing... Wouldn't it suffice to say the equality for two events ? Can't the above equality be derived from the equation considering just 2 Events? In fact, by Occam's Razor, isn't the latter the way it is supposed to be ? Rkr1991 (Wanna chat?) 13:43, 11 September 2009 (UTC)

Stating it for two events implies the equality only for finitely many events ("finite additivity"), not for countably many ("countable additivity"). It is in fact possible to reject countable additivity and get a different kind of probability theory, but it's not common. Shreevatsa (talk) 14:03, 11 September 2009 (UTC)
Why woldn't it hold for countable additivity ? Can you please explain ? Rkr1991 (Wanna chat?) 04:13, 12 September 2009 (UTC)
Why would it hold? :-) It is clear how to extend it from 2 to any finite number (by induction etc.), but it's not possible to extend it to infinitely many. [BTW if a mathematical discussion gets too far from discussing the article itself, it may be best to take it to the reference desk.) Shreevatsa (talk) 04:26, 12 September 2009 (UTC)

## Page name

I think the name of this page is misleading, since there are other ways to formalise probability theory than Kolmogorov's (one example is Cox's approach, linked at the bottom of the article). It should be called something like "Kolmogorov's axioms for probability theory" instead. Nathaniel Virgo (talk) 13:34, 14 April 2011 (UTC)

## Re-organizing First Axiom?

So...maybe I'm a horrible person, but I've been reading this page for a while and just noticed that axiom 1 kind of spells out

${\displaystyle P(E)\in RAPE\geq 0}$

Feel free to disregard if it's just me...but maybe a simple rotation of the P(E) and R terms would help...? 71.197.0.228 (talk) 06:51, 2 May 2012 (UTC)

## Errors in these axioms

When a reader carefully studies the sections First axiom, Second axiom, and Third axiom (the heart of this article), should he/she understand these to be building on the second paragraph of the introdoction (the paragraph that begins "These assumptions can be summarised as....")? Or are they meant to be independent of that paragraph?

The reason I ask is that First axiom defines F, which has already been defined in the second paragraph. This suggests that they are meant to be independent. However, Second axiom never defines Ω. There is no indication that Ω has any relation to any of the symbols yet mentioned, nor even that Ω is a set at all. Then to the reader's surprise, a set subtraction operation Ω\E is performed later on, suggesting that Ω is a set.

Moreover, the Consequences section is wrong, or the Axioms are misstated. Consider the following example. Let F = {a, b, c}. Let Ω = {a, b}. Let P({a}) = 0.5, P({b}) = 0.5, and P({c}) = 0.5, and assume that P is fully additive. You can easily verify that this example satisfies all three axioms: in particular, every subset of F has non-negative finite probability, and P(Ω) = P({a, b}) = 1.

Yet it does not satisfy the so-called numeric bound consequence, because there is a subset of F whose probability is greater than one: specifically, P({a, b, c}) = 1.5.

One way to fix this is to explicitly state that Ω is just another name for F. — Lawrence King (talk) 21:46, 9 November 2012 (UTC)

Another way to fix this issue would be to use the same symbol Ω or ${\displaystyle {\mathcal {F}}}$ throughout the article. In addition, I also noticed that it might be possible to change the first axiom to read ${\displaystyle P\left(E\right)\in \left\{x\in \mathbb {R} :x\geq 0\right\}\,\forall E\in {\mathcal {F}}}$ instead of what it reads now.
RandomDSdevel (talk) 01:15, 1 April 2013 (UTC)
P.S.: Oh, wait: don't fix it! I just remembered that ${\displaystyle {\mathcal {F}}}$ is the set of all subsets (or, alternatively, all members of the power set ${\displaystyle {\mathcal {P}}\,\left(\Omega \right)}$) of the sample space Ω to which one can reasonably apply the probability measure P! For finite sample spaces, ${\displaystyle {\mathcal {F}}}$ would contain all subsets – i.e.: all members of the power set ${\displaystyle {\mathcal {P}}\,\left(\Omega \right)}$ – of the sample space Ω, but one has to limit the σ-algebra ${\displaystyle {\mathcal {F}}}$ to include only the measurable subsets of the sample space Ω when it is infinite.