Talk:Almost surely

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[edit] Dart example

I just came across this article and I like it. I think it would be helpful for the example of hitting the diagonal of a square with a dart to point out that the setting described is very idealized in that it depends entirely on an idealized notion of space and spatial measurement: the width of the diagonal is assumed to be 0, and the width of the tip of the dart is likewise assumed to be 0, and the precision of determining where exactly the dart hit is assumed to be arbitrarily precise (i.e. infinite or zero); of course this is not the situation anybody could realize with a physical dart thrown at a physical square with a physically materialized diagonal. Almost surely the authors of this article are very much aware of this situation, but almost as surely some of the more unwary readers of the article are not. — Nol Aders 23:19, 26 December 2005 (UTC)

I tried a parenthetical comment there. If you can think of a better way to say it, or if you have anything else to add, feel free to edit the article yourself! -Grick(^{talk to me!}) 08:00, 27 December 2005 (UTC)

Perhaps we should change the dart example entirely, to something more obviously numerical in nature? For example, how about picking an arbitrary number between 1 and 10, and the odds of it being exactly pi (rather than, say, any other irrational or rational number)? Scott Ritchie 21:41, 15 January 2006 (UTC)

I would just like to agree that this is a good article. Alot of the mathematics articles are hard to understand, but this one is interesting and accessible. Cheers —The preceding unsigned comment was added by 134.10.121.32 (talk • contribs).

I think there is a flaw here. The article assumes that space is not quantized thereby allowing the diagonal to have a zero area, and at the same time it assumes that space is quantized since is allows the possibility of a zero point dart landing exactly on it. I think I can prove that if space can be divided infinitely, then the dart can in fact, never land right on the diagonal. I have only a rude training in Mathematics, so if any mathematician has a comment on my statement, it would be very enlightening.59.144.147.210 17:43, 15 November 2006 (UTC) Bhagwad

The dart will certainly land somewhere, right? Well, what is so special about the diagonal that sets it apart from the points you assume the dart can hit? There is no point in the square that is off limits or impossible to hit. You have to keep in mind that "impossible" and "probability 0" are not the same thing. "Probability 0" things happen all the time (like if you mapped your exact route to work today with infinite precision) but will almost surely never happen again. On the other hand, "impossible" events cannot happen at all. (Now, you could argue that some physicists have evidenced that space and time could be in fact discrete; and then you could argue that anything we can physically create is finite in every sense, and that the only infinite things are abstract mathematical notions.... then "almost sure" and "sure" would be equivalent... but that's not near the fun!) - grubber 19:48, 15 November 2006 (UTC)

Hmm. You're right. 59.144.147.210 04:26, 16 November 2006 (UTC) Bhagwad

[edit] No trouble at all with finite sets

I think it should be mentioned on this page that the need for perplexing terminology only arises if probability is defined as the limit of frequency. However, there is no need to define probability in terms of infinite sets (cf. Cox's derivation of probability theory, in his book "The Algebra of Probable Inference"). Given a finite set of propositions, probability 0 always implies a false proposition ("an impossible event" in your terms) and vice versa, and probability 1 always implies a true proposition ("a certain event"). If you wish to consider what happens with probabilities when a set of propositions (events) becomes infinite, you should pass to the limit in a well-defined fashion. "Well-defined fashion" requires specifying the operation by which you extend the originally finite set to approach infinity. Better yet, restrict yourself to finite sets of propositions in your applications and avoid the need for metaphysical terminology altogether.

For a thorough (but unfortunately difficult to understand) discussion of paradoxes which arise from the overeager introduction of infinite sets into considerations of probability, I refer you to Chapter 15 of Jaynes's book [1]. Jploski 14:39, 11 February 2007 (UTC)

I'm not quite sure what your point is. But in any case the standard, infinite set axiomatics of probability are quite good if you want foundations for things like Brownian motion. Charles Matthews 16:01, 11 February 2007 (UTC)

Two comments: Almost sure is a concept that is valid whether you define probabilities based on "limits of frequency" or from a purely mathematical/topological viewpoint. Second, "probability 0" and "impossible" are synonymous in countably infinite sets as well as finite. The issue only arises when we have a space that is larger. - grubber 17:22, 11 February 2007 (UTC)

[edit] Preferred version

I like http://en.wikipedia.org/w/index.php?title=Almost_surely&oldid=101537841 better than the current version, don't care for the huge change. Lilgeopch81 20:30, 12 February 2007 (UTC)