Talk:Almost surely

message: The given definition is just non-sense.   Citation: "pn > \tfrac{(1+\epsilon) \ln n}{n}" where the right hand side tends to zero as n -> \infinity (Did I get this right, asymptotically refers to n -> \infinity ?). A probability greater than zero we have always, therefore the non-sense. But I do not know how to improve this section.  — Preceding unsigned comment added by 192.41.132.103 (talk) 09:13, 13 August 2014 (UTC)
thread_title: asymptotically almost surely

message: 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 . Jploski 14:39, 11 February 2007 (UTC)
replies: text: 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)

text: 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)
replies: text:  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)
replies: text:  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)
replies: text:  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)
thread_title: No trouble at all with finite sets

message: 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)
thread_title:  Preferred version 

message: "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.
: Reply to above: it's not quite valid to estimate the total number of primes by estimating their "proportion" in the set of intergers, and by furthermore seeing what that fraction tends to as the total set tends to infinity. A similar process would lead to the erroneous conclusion that the set of integers is twice as large as the set of even integers, when in fact their cardinality is the same. Or, for that matter, that the set of integers smaller than 10 (or smaller than a googol, or whatever) has a cardinality of zero. These sorts of seeming paradoxes aren't the result of "almost surely" anyway. 72.72.210.38 (talk) 17:19, 12 October 2011 (UTC)
replies: text:  "Almost surely, a given number is not prime." is the true statement that I think was intended. Treedel (talk) 23:55, 12 October 2011 (UTC)
thread_title:  Almost surely there are no primes 

message: 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)
replies: text: The probability of getting 500000 times heads and 500000 times tails is definitely way larger than 2^-1000000. The probability of getting first 500000 times heads and then 500000 times tails is 2^-1000000 (and the same, of course, for any other specified order). The total probability of getting 500000 of each, in any order, is \textstyle {1000000 \choose 500000}\, 2^{-1000000} (see binomial distribution). 85.226.204.72 (talk) 10:55, 7 July 2010 (UTC)
thread_title:  Is this a correct explanation? 

message: 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)
replies: text: Can you (or someone else) please elaborate on how this meaning arises? It's not clear in the article why this distinction is necessary, in practice. In what cases would it have any pratical implications (other than some math teacher getting upset) to state that an event happens "surely" when its probability is one? --217.157.165.109 (talk) 15:34, 31 August 2011 (UTC)
thread_title:  Meaning in practice 

message: In the section "Tossing a coin" it says that "1/infinity = 0". I'm not a mathematician, but isn't division by infinity impossible? Shouldn't it be  \lim_{x \to \infty}1/x = 0? --Jak86 (talk) 04:20, 6 December 2008 (UTC)
replies: text: Whether or not it's impossible is really a question of definitions. Defining 1/infinity to be 0 typically creates no problems, and it's done in the extended real number line, the real projective line or the Riemann sphere. (Defining it to be anything but 0 would be begging for trouble.) -- Jao (talk) 18:43, 12 January 2009 (UTC)

text: I agree with Jao.  Another thing, and I don't how correct this is.  In physics, when I see an \infty in some equation, I often assume that it implying a limit.  So, reading 1/\infty, I would have read that as being what you suggested it ought to be.
Monsterman222 (talk) 10:45, 14 February 2014 (UTC)

text: What you really can't is subtract one infinity from another, or divide it by another, or divide 0 by 0 (if not as a value of a function within a limit expression of there's the possibilty of applying l'Hôpital). Otherwise, using 1/0 = infty and 1/infty = 0 is unproblematic and often done.--131.159.0.47 (talk) 19:45, 10 July 2014 (UTC)
thread_title:  Possible mistake? 

message: Regarding
:"If an event is sure, then it will always happen. No other event (even events with probability 1) can possibly occur." (my emphasis),
This sounds to me like it's possible to have two mutually exclusive events A and B such that A is sure to happen and P(B) = 1. I don't have to be a measure theory expert to see that that's impossible. Should we simply remove that confusing parenthesis? —JAO • T • C 10:02, 13 February 2009 (UTC)
replies: text: It may seem implausible, but the fact of having mutually exclusive events each of which have probability 1 is a direct consequence of the convention of assigning probability 1 to both events that are sure to happen and events which will almost surely happen. For example, suppose there is an infinite lottery to take place, which will be done by selling an infinite number of tickets to different people, such that there is one lottery ticket for each positive integer. After distributed all those tickets, one ticket will surely be drawn and declared the winner. For any particular ticket, the probability of that ticket being the unique winner chosen once a selection is made at random from a uniform distribution is 0. Thus the probability of at least one ticket being picked is equal to the limit as x goes to infinity of the summation from i = 1 to x of 0, which sums to zero. The complement gives the probability that no ticket will be picked as 1. So a ticket is sure to be picked, but almost surely no ticket will be picked. So if E is the event of at least one ticket being a winner, P(E) = 1 and P(not E) = 1. —Dromioofephesus (talk) 19:41, 5 April 2018 (UTC)
thread_title:  "(even events with probability 1)" 

message: I'm no mathematician, and I'm not sure about the throwing of the dart and the almost sure result that it hits the target. The target cannot be the only thing in that universe, as if it were, there would no no dart, no-one to throw the dart, and no oche for the thrower to stand at (if he/she existed anyway). Given the necessity for these additions to the target's universe, one finds that there is a possibility that when the dart is thrown, it is impelled with insufficient force to reach the board. (Even happens in championships - but almost never...) I realise that this is an example for the masses, but it worries me (a little). Peridon (talk) 17:03, 26 September 2011 (UTC)
replies: text: The premise that the dart must hit the board is no more unreal than the premise that the dart has a one-dimensional head, and thus hits the board at exactly one point. If we permit the dart to have a cross-sectional area, even one as large as an iron atom, then the chance of including a specific point becomes ε, with σ being the cross-sectional area of the dart. Treedel (talk) 00:04, 13 October 2011 (UTC)
replies: text: I also take issue with the dart analogy. What's to stop the thrower directing the dart /away/ from the board? What's to stop the thrower throwing the dart into his foot?  — Preceding unsigned comment added by 61.14.115.68 (talk) 00:31, 30 January 2013 (UTC)
thread_title:  Back to the flight of the dart 

message: Low-importance statistics articles?
https://en.wikipedia.org/wiki/Category:Low-importance_Statistics_articles 
This is an astounding way (a bad way) to classify this article. Anytime that one defines a continuous probability density (a distribution), this instantly brings about the possibility of events of zero probability. Among other things, it follows directly from the properties of the Riemann integral.47.215.211.115 (talk) 11:14, 12 January 2017 (UTC)
thread_title:  Low-importance statistics articles? 

message: A significant portion of the current article is dedicated, essentially, to defending the historical choice of the terminology "almost surely" against the possible view that "surely" would have been better choice of terminology for the same concept.
This issue has reasonable arguments to be made on both sides, but it doesn't seem like a suitable task for the wiki.  I think examples like the square diagonal should be given, but only to illustrate the mathematical definition, as opposed to (as is the case currently) being used to argue that these scenarios are better described in English by the words "almost surely" instead of just "surely".
In any case, as it currently stands, I believe the sections constitute original research, since they do not cite sources discussing the merits of using "almost surely" or "surely" as the name for this concept.  The article as it currently stands may also mislead readers into thinking that the word "surely" has a separate commonly used mathematical definition (which supposedly means about an event that it is equal to the sample space).   Discrete (talk) 14:09, 24 April 2017 (UTC)
replies: text: I have changed to title for this section and cleaned it up a bit. Discrete (talk) 18:22, 24 April 2017 (UTC)
thread_title:  Almost surely vs surely 

message: I propose to delete this section.  A probability P(N)=0 cannot be assigned—as has been done in this section—to every integer N \in \mathcal{N}, because then \sum_{\mathcal{N}}P(N) = 0.  But, by the axiom of unitarity, P(\mathcal{N})=1. Hence, P(N) cannot be 0 for all N \in \mathcal{N}.--Jt512 (talk) 06:28, 14 November 2017 (UTC)
replies: text: I agree. Another reason is that there is a finite upper bound beyond which a number is too long to be denoted, in any system, during a human's life time. (I naively assume that "guessing" includes denoting in some form, while e.g. dart throwing does not.) - Jochen Burghardt (talk) 10:57, 14 November 2017 (UTC)

text: I have deleted the section accordingly. --Jt512 (talk) 23:42, 14 November 2017 (UTC)
thread_title:  Selecting a random number 

message: The remark In probability experiments on a finite sample space, there is no difference between almost surely and surely. is misleading. It suggests (incorrectly) that in a probability space (\Omega,\mathcal{F},P) with finite \Omega a probability of 1 means that the event equals \Omega (or a probability of 0 means that the event equals \emptyset). There is nothing in the axioms of a probability space
which requires this. One can come up with this misleading sentence when viewing probability space only via the special case of Laplace experiments, which is was too limiting.  — Preceding unsigned comment added by 139.30.119.241 (talk) 21:52, 2 November 2018 (UTC)
thread_title:  Remark on finite sets incorrect 

message: Is there a noun form of the phrase "almost surely"?  It's tempting to use the phrase "almost certainty", contrasting with "absolute certainty".  But "almost certainty" isn't exactly grammatical.... — Smjg (talk) 18:26, 11 August 2019 (UTC)
replies: text: I never saw such a noun form, neither in German (my native language), nor in English. - Jochen Burghardt (talk) 08:22, 14 August 2019 (UTC)
replies: text:  I've seen no such usage attested. If you're asking whether there could be such a form, yes of course there could -- such as in the variant 'near-certainty'...near and almost are both casual ways to explain the concept in a way that appeals to intuition while being technically inaccurate: you wouldn't expect someone who says they are "almost 50 years old" to be exactly 50 minus an infinitesimal fraction of a second, but you can certainly imagine such a situation. Thus it is with the name of this concept. Arlo James Barnes 01:25, 10 February 2022 (UTC)
thread_title:  Noun form 

message: If an event will almost surely happen, does it mean it will eventually happen? Regardless of how close you get to a probability if 1, there’s always the possibility that it doesn’t happen when it can happen. --Heymid (contribs) 20:59, 16 April 2021 (UTC)
replies: text: You grasp the crux of the term, I think. It's delinking probability and inevitability. An event that has probability one (not just close to one but all the way there) can be infinitely unlikely to not happen, but that doesn't mean it's inevitable; only occurrence / nonoccurence over time can 'reify' the probability. Arlo James Barnes 01:28, 10 February 2022 (UTC)
thread_title:  Almost surely? 

message: An editor has identified a potential problem with the redirect :Wp1 and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 9#Wp1 until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  signed, Rosguill talk 19:20, 9 February 2022 (UTC)
thread_title:  "Wp1" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: I'm sorry, but this is an embarrassingly stupid article, and unfortunately an illustration of the key flaw of wikipedia, where self-proclaimed "experts" write nonsense articles. Most real experts find it frustrating and a waste of time trying to stem the tide of nonsense, and are too busy doing real work to engage in the needed edit wars against the false consensus. So I will leave this comment and perhaps the editors of this article will correct the article, preferably to consist of a brief definition with a few brief examples.
Now, I believe the editors of the article would agree that "surely" (same as "certainly") means 100% probability, means something we are 100% sure about. If you think "almost surely" means 100% then I would hope you would agree that "surely" means 100%. Right?
Moving on to "almost surely", the key word "almost" means that the probability is *almost* (same as "not quite") 100%, but *not* 100%, such as 99.9% probability. If it was 100%, as the article falsely claims, the "almost" would be omitted. Look in any dictionary: "The work is almost done" does *not* mean the work is done, and "We are almost home" does *not* mean we are home. If you want to change the meaning of the word "almost", as the article does, then there is no discussion, because you are not free to define commonly used words as you choose. The upshot: The Emperor has no clothes. The article is nonsense.
The pitiful long-winded "examples" in the article fall flat upon examination. In contrast, there are countless things we can be sure (meaning 100% sure) about, and countless things we can be "almost sure" about. For example, I am "sure" that two goats plus two goats is four goats. I am "almost sure" my car is in the driveway, even though I have not looked for 10 minutes. Who knows, maybe it got towed away while I was not looking. So there is a real and definite difference between "sure" and "almost sure", regardless of the nonsense spouted in the article. 97.113.102.52 (talk) 21:07, 2 December 2022 (UTC)
replies: text: Like it or not, "almost surely" is mathematical jargon in probability and measure theory; the article does decscribe its meaning adequately. Your arguments about goats and cars apparently refer to the common-language meaning of "almost surely", which is (here I agree with you) quite different. However, the latter meaning hardly deserves an own article. - Jochen Burghardt (talk) 21:27, 2 December 2022 (UTC)
thread_title:  Embarrassingly stupid article