Talk:Almost surely

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 agree! Paul Haymon 10:24, 12 April 2007 (UTC)

I agree as well - very informative and enlightening.

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

I added a paragraph in an attempt to really drive this point home. Floorsheim (talk) 00:12, 4 March 2008 (UTC)

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)

I think the source of the confusion is mathematical rigour. This article, and many others on Wikipedia, is an intuitive explanation of very rigorous mathematical constructs; in this case, of something called a probability space. As such, this article provides intuitive insights to aid in the understanding of the concept, but ironically this article is quite immune from intuition. Every phrase in any mathematical article has an absolutely precise, non-vague meaning (at least they can be translated to predicate sentences in ZFC). This include the phrase "almost sure." Unfortunately, "almost sure" already has an English, "normal" meaning, and so discussions can turn metaphysical without warning. At this point, those who are using the precise mathmatical definition are talking about something entirely different than those who are using the "real-life," English meaning. What's worse, phrases like "infinitely thin" and "infinity" have no mathematical meaning until, well, we define them. And there are many pre-existing definitions that vary wildly from context to context (but every one of these definitions is precise and non-vague). Though we can give them "real-life" meanings, it would avoid the entire point of mathematical rigour and, unfortunately, the aim of this and all other mathematics articles. Though it may be argued that these articles bring mathematics to the wider public, and hence "real-life" meanings are therefore to be encouraged, it is wrong to conclude that these meanings are the mathematical definitions. It is dangerous to give the wider public the impression that mathematics is a vague subject and statements can fall into grey areas. Mathematics is an exact and absolutely precise language, discipline, and form of creativity. Even the undecidability results are rigorously proven. And come to think of it, this is absolutely a good thing. It's counterintuitive results like these that make mathematics beautiful---things that we don't see when we use our intuition. Perhaps these events that "almost surely" occur are some of those things. - weixifan 23:21, 29 March 2007 (UTC)

I think the article does a decent job of offering both intuition and rigor. Can you give an example of something rigorous that this article is missing? - grubber 04:28, 30 March 2007 (UTC)

Personally, I'd like it if the article mentioned that the probability of the dart hitting any specific point is zero, yet the dart obviously does land on a specific point. Eoseth 15:33, 14 April 2007 (UTC)

Preferred version

I like http://en.wikipedia.org/enwiki/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)

Almost surely there are no primes

"Almost surely" may produce misleading statements. For example, the fraction of prime numbers tends to 0 as the number of numbers under consideration goes to infinity. When considering all possible numbers then, the fraction prime is almost surely zero while initially the fraction of primes is nearly unity. The set of integers is not necessarily a good measure for the number of primes.

"Almost sure" versus "sure"

This section currently reads as follows:

The difference between an event being almost sure and sure is the same as the subtle difference between something happening with probability 1 and happening always. If an event is sure, then it will always happen. No other event (even events with probability 0) can possibly occur. If an event is almost sure, then there are other events that could happen, but they happen almost never, that is with probability 0.

This wording seems to suggest a sure event is only found in a sigma-algebra which consists only of the empty set and the sample space. Is that what's intended here? --Mark H Wilkinson ^{(t, c)} 17:46, 13 September 2007 (UTC)

On 'Tossing a coin', a coin can land on it's edge. Heads or tails is far from a sure event, maybe not even 'almost sure', if one considers this article: [2]. —Preceding unsigned comment added by 75.15.205.162 (talk) 11:40, 25 September 2008 (UTC)

I believe darts sometimes miss dartboards as well. I'm afraid I don't have a link to an Ivy League website to back that up. H.G. 12:06, 25 September 2008 (UTC)

No mention of limits

There is a lot of talk of things being zero, rather than limiting to zero, which seems to be the crux of this issue, is there something that i am missing? Sure seems to be the concept of a zero probability, whereas almost sure seems to be the concept of a probability limiting to zero 129.78.64.101 12:07, 24 September 2007 (UTC)

Not really. Probability zero doesn't mean 'cannot happen': that's the essential. Charles Matthews 12:35, 24 September 2007 (UTC)

Is this a correct explanation?

The dart one is fine and good, but I've been thinking of an explanation for the difference between "surely" and "almost surely" that will be even clearer, and I think I've got one. Can someone tell me if this is mathematically correct?

Consider an operation, F that has only one possible outcome, A. Operation F will surely result in A, no other result is possible. Now consider an alternative operation, G, that has two possible outcomes, A and B, but that outcome A occurs with 100% probability. In this case, G will almost surely result in A (or almost surely not result in B), because although it happens with 100% probability, another result is possible.

My questions are a) is this correct and b) can it be made even clearer? Sloverlord (talk) 14:52, 12 May 2008 (UTC)

It is definitely correct. The best interpretation of what it means for an event E in a sample space X (if you like you can assume X is a measure space), is that the measure of the complement of E (or the probability of the complementary event X-E) is 0 but E is a proper subset of X. Or in other words, the probability that E will not happen is 0, but it is possible that E can 'not happen'. Your example is perfectly fine and I think that it is quite clear. However, I think that the example given in the article is the best illustration of the definition but your one is good too.

Topology Expert (talk) 08:43, 15 September 2008 (UTC)

Meaning in practice

The practical meaning of almost surely is: If E is a potential future event and it will almost surely happen and one has explicitly asked the question whether E will occur, then it will occur. JRSpriggs (talk) 05:39, 26 May 2008 (UTC)