2963:
little about what we are reading about. Aside from the ambiguity, the rest of the sentence is extremely confusing as well. Although I am someone who has studied probability not only for my college degree, but also on my own to further my knowledge, I still had to read that sentence over and over again to understand what the writer meant. I would not consider probability an attitude of mind, because this simply does not make sense. The “proposition of interest” and “attitude of mind” are unnecessary and confusing when defining probability. Probability is the analysis of the outcome of an uncertain event. The given
Knowledge definition gives several examples of what probability is, rather than defining it. It states that, “The proposition of interest is usually of the form…” This is not only an example instead of a definition, but it is an extremely poor example. The word usually is very vague, leading the reader to believe that the given example does not apply to probability all of the time. The definition also ends with probability in an applied sense. But what does that really mean? Because there are no other examples of another sense that probability applies to, this makes the definition unclear. The definition then goes on the describe probability in mathematical terms using the range between 0 and 1. For anyone who has not dealt with probability in a mathematical sense, this definition most likely makes little sense. The way it is written is very vague, but what it is trying to say is that there is a range of outcomes in probability starting from 0 (the event will definitely not occur), to 1 (the event will definitely occur). These ranges are also often written in percent form, 0% to 100%. I believe that by adding the fact that these numbers are percentages, the numbers already start to make a more sense and become more valid. The range between these numbers is the percent of likeliness of an outcome. While this definition gave many examples that do not help the reader to further understand probability, I would perhaps include one common example when dealing with the mathematical sense, even though definitions should not include examples. For example, the probability of a fair coin landing heads after it is tossed is .5, or 50%. This is because there are two possible outcomes, heads or tails, and both are equally likely. If the coin had been unfair with heads on both sides, the probability of landing heads would be 1, or 100%, since there are no other possible outcomes. As stated in the original definition, “The higher the probability of an event, the more certain we are that the event will occur.”
2063:
is in a dynamic equilibrium in which all events lead from prior events in a never-ending chain. Consequently, if you buy his argument which I do, then the probability of anything occuring is 0 or 1. This leads me to state that probability does not apply to the universe but to the observer who wants to predict the future with limited data. For instance, weather forecasters are typically right 50% of the time. Whatever data they use gives them a 50% PROBABILITY of being correct. If they knew everything there is to know their predictions would always be 100% correct. Anyway, another interpretation of probability has nothing to do with 'events' in the world, and everything to do with the person attempting to predict future events. Cheers. --
237:, and "0" to mean "impossibility", by definition, without taking any philosophical position on what those terms mean. The ordinary interpretation of those terms to ordinary circumstances of life (like rolling dice) is entirely obvious, useful, and clear. The fact that some philosophers argue about it is a subject for some other article; I, and our readers, are well-served in their ordinary lives by the simple understanding that the probability of drawing the 17 of hearts from a deck of cards is 0, and the probability of drawing a card with two sides is 1. If you want to enlighten them with some deeper understanding, write about it and put link here. Until then, let's keep the text here practical and useful. --
2837:
probability of one particular real number in uniform continuous distribution in the segment therefore would intuitively be one over infinity -- expressed as a limit this tends towards zero as the number of points approaches infinity. This changes when you consider a range (a set of points), rather than an individual point. A range in the uniform distribution, for example, could be the interval from zero to .25, which of course would have a probability of 25%. In physical space, lengths are not infinitely divisible, eventually you reach mechanical, molecular, atomic and quantum limits.
257:
10! So, you could have drawn 4.1234234156839201248324234... How many of these numbers are there? Infinite. So the probability that you draw exactly the one I wrote is zero, i.e. it's almost impossible. But it's still possible! Thus P=0 means "almost impossible" rather than impossible. The misconception that P=0 means impossibility comes from the "rule" that P(empty set)=0, which means that an impossible event has zero probability, i.e. the inverse implication. The latter is true, the former is not. At least for infinite sets, no matter whether they are countable or not.
1928:
exists, it can never ALL-Not-exist at any time, otherwise it can never exist (all-not-exist). So that means all-potential exists (probabilities even), must exist in even in partial form, in some way. Otherwise probability calculations would be impossible (to all-not-exist, is to never be possible to occur or exist). If you make a claim my statement is unintelligable, you make a claim to know which words and which statements, else you don't have a claim to know, so if I am in error, please point out which words and which statements, so that I can correct myself.
2968:= 0, and P(S) = 1.” Glenn Shaffer explains these axioms in a more wordy, less mathematical terms for the average reader to understand. First, the probability of an event must be greater than or equal to zero. If it is less than zero, then the event is not possible. Lastly, Shaffer states that the probability that the event will produce any of the available outcomes is 100%. Going back to the coin example, the probability that a fair coin will land either a head or a tail, is 100%, because those are all of the available outcomes.
334:
notions of probability. So whatever you prove from the axioms of probability theory theory will then apply to the long-term relative frequency notion of probability as well. So a question such as "if you flip a fair coin 5 times, how likely is it that you get at least 4 heads", initially using the intuitive probability notion, can be translated in the language of mathematical probability and solved there. If mathematical probability and everyday probability had nothing to do with each other, then the latter would be useless.
31:
391:"Probability in mathematics" -- identifying probability with coin flipping is extremely limiting. The law of large numbers is a theorem. The limit definition is a feature of a particular school of thought (frequentism). I like "Rosencrantz & Guildenstern" but the coin flipping scenario from R&G doesn't shed any light on "Probability in mathematics. The mention of Bayesianism was clearly an afterthought.
136:
intended meaning is in common use, while still leaving open the possibility of more advanced interpretations that other articles can cover. Furthermore, the common usage is not at all "incorrect" as you imply--it's completely correct in most of the contexts in which probability is used. It I am "drawing dead" in a poker game, that means I have calculated the probability of winning as 0, which means
1471:
system? (eg. air molecules in a room - the highest probability outcome is the one in which the number of atoms is about equal per unit volume; another way of saying this is that it is entirely possible that all the air molecules at once could shift to one corner of the room but the length of time required to observe for this low-probability outcome is longer than some absurdly high number). --
5333:, etc.). This is the article about the basic concept of probability, including prominently the non-mathematical ideas and interpretations that are referred to by this term. It's not reasonable or helpful for more than a third of the article to be taken up by equations that are already explained more clearly and thoroughly in the articles that actually treat those topics. Let's use
1988:"Probability is the chance that something will happen or be the case" would be better, stripping away one of the probability-laden operators. 3. The word "chance" has to my ear OBJECTIVE connotations. But 'probability' can be a subjective notion: degree of belief. "Chance" seems to fit awkwardly with that notion. But thanks for taking the trouble to write this article!
2959:
we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the confidence a person has that a (random)event will occur.
5503:
that good luck would only be discribed in the short term, as probabilities usually even itself out in the long run. I mean if you hit the 35-1 odds on the roulette table, but was your 38th time you played and never hit, that would not be lucky. However if you hit the same number 38 times in a row on a 35-1 shot, that would be extremely lucky. 16 Jan 2007
155:
expressed and how they relate. That information is more knowledge about language than about math, but again I see no reason why a page about "probability" should be limited to strict mathematics. Perhaps there is a better way to word the above to be more rigorous and also useful to a lay audience, but I can't think of it off hand. --
2967:
NJIT. Both of their main mathematical focuses aim a probability, making them two of the best sources for this proposal. From a mathematical standpoint, Richard
Sheaffer states that “Probability is a numerically valued function that assigns a number P(A) to every event A so that the following axioms hold: P(A) : -->
2828:"on the interval , every point has the same probability, which must be 0" Why must it be 0? Sorry I am not an expert in maths. Also I am failing to understand why this is wrong. "If probability is equal to 1 then that event is certain to happen and if the probability is 0 then that event will never occur."
4879:
99:"certain" and "impossible" events have no measure, and we cannot speak of them within probability theory. If this seems strange, it's related to the notion of denumerability (countability). Denumerable sets have measure zero. The easiest example of a denumerable set is the set of all rational numbers.
2892:
loosely speaking the probability of an event is the number of positive outcomes divided by the number of all possible outcomes. Throw a dice: probability of getting a 3 is 1/6. But now consider something where you have an infinite number of possible outcomes: pick a random number between 3 and 4, the
2836:
particularly the sentence, "Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space."). So, the
2225:
I'm guessing it might refer to his finding that for three observations with identically distributed errors around some value, the median of the posterior distribution of that value (under a uniform prior) minimises the mean absolute error. (See for example p. 213 in the translator's afterword to this
2082:
In the last year more than 100 editors have edited the article (also counting non-vandal editors whose edits were reverted, like yours), so it is very much the result of a collaborative effort, and it isn't quite clear what you mean by "running
Probability". Your edits were reverted because they were
1481:
Yes I think it would be a good idea to explain the difference between how the physicist and the mathematician view this. In essence, a physicist is someone who would consider an event with a very very small probability (but larger than zero) as totally impossible, while a mathematician is someone who
333:
of the axiomatic system in the real world. For instance, the long-term relative frequencies often used in everyday life are a model of the probability axioms (or so we believe: this is a statement outside of mathematics and cannot be proven); the probability axioms were constructed to model intuitive
311:
This is not a trivial issue but has enormous impact on people's (even many physicists') incomprehension of probability as it is used in physics. In particular, the notion of "probability" used by the
Copenhagen interpretation of QM is literal nonsense. It's an intuition about the everyday world which
198:
The beauty of it all is that the dogma is "mathematical truth". One starts with definitions, then develops a theory standing on those definitions. I hold very tightly to this dogma. There may be academics from a wide variety of fields that may dispute the origin, foundations and claims of probability
94:
That's the whole problem. What the reader "knows", versus what the reader thinks he or she knows, it not the same here. (I just read your peer-review comments, so you can't slither away on a "common sense" argument.) The problem really is in the fact that probability is based on the notion of measure
5502:
Is there enough to say on luck? I have tried to look for a specific definition for luck, but cannot get one in the context of probability. All i can think of it that luck is for an unlikely favourable outcome(although bad luck is an unlikely unfavourable outcome). I would also mention sometime about
2867:
and what exactly does the last part fo this mean. "An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true" What are the opposites here? A possible event doesnt have a probability of 0, and a uncertain event isnt 1. Surely
2062:
First, I'd like to reiterate that whoever is running
Probability is too focused on the math, which for the majority of readers is a dry side issue. What is really interesting to many (most?) people is what does probability mean to us living in the world. I think Pierre Laplace said that the universe
1223:
In other words, by saying that "the probability of heads is 1/2", this law asserts that if we flip our coin often enough, it becomes more and more likely that the number of heads over the number of total flips will become arbitrarily close to 1/2. Unfortunately, this theorem says that the ratio will
738:
In other words, by saying that "the probability of heads is 1/2", this law asserts that if we flip our coin often enough, it becomes more and more likely that the number of heads over the number of total flips will become arbitrarily close to 1/2. Unfortunately, this theorem says that the ratio will
324:
Regular stats courses usually don't give the sigma-algebra defintion of probability, but courses called "Probability and
Statistics" or "Stochastics" do. I disagree with your statement that mathematical probability has nothing to do with everyday notions of probability. It's an axiomatic system, and
3139:
I'm moving the huge mass of (rather sloppy and unclear) errata on mathematical properties of probabilities out of the article and storing it here. I don't believe this stuff has a place in a top-level, non-mathematical summary article; I've replaced it with a brief summary of the basic concepts and
2962:
The opening sentence brings great concern as to what probability really is. It begins with, “Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth is not certain.” The word ordinarily is not only ambiguous, but makes the opening sentence tell us very
2958:
The given
Knowledge definition is as follows: Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth is not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are
2951:
In this proposal, I will discuss why the
Knowledge definition for “Probability,” is not a good, valid definition, and needs to be changed. Probability is something that is used every day. It is what helps us further understand the outcomes of the events in life that are uncertain. However, in order
2904:
The article currently reads "The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty)." I think we've established in the talk section that this statement is incorrect. Based on the talk page,
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page into a good article. The paragraphs above should fit nicely into an improved probability interpretations page. That is in fact also true for a large part of the current article. It should be moved to the interpretations page and removed from this page. I'm not sure what would be appropriate to
974:
talk page. I would like someone who feels they can be independent and fair to both sides. This page includes religious discussions and is somewhat controversial, but I don't think this should affect the specific discussion regarding the probability of an event. You can also leave a message on my
256:
Here is a practical example that shows how that "almost" does have an effect on reality, it's neither epistemology nor metaphysics... Draw a number between 1 and 10. What is the probability of drawing one particular number? 1/10? Wrong. Because I was talking about an irrational number between 1 and
98:
In a nutshell, the probability of any particular event, including taking the values 0 or 1, is 0. That is, it forms a set of no measure. For a concrete example, integrating f between a and a equals 0. Which is how random variables assign values, through the action of the associated integral. So the
5478:
The statement that, "One in a thousand probability played a thousand times has only a one in a million chance of failure," is patently false. An event with a .001 probability, repeated one thousand times, independently, has a 36.7% chance of not occurring in those thousand trials. That's a far cry
2574:
I was always taught that it's not meaningful to assign probabilities to things which have occurred, except 0 or 1, since they either definitely did or definitely did not occur, and ignorance of which is not a basis for a probabilistic statement. To give such a past event a probability of 0.5 would
177:
Oh yeah... common sense would say that P(E) = 0 means impossible, etc., but that doesn't lead to useful definitions for a theory of probability (if it did, the introductory texts would teach it that way instead of studiously avoiding the topic). I am really hoping someone currently teaching a grad
169:
I agree with LDC that the page should be accessible to lay audiences, but let's not insult their intelligence. Just because the distinctions are subtle (subtile heh) is no reason to hide them, and these fine points really are part of the story of how and why probability works, in both the pure and
135:
Yes, I think the overly careful avoidance of "certainty" and "impossibility" here is more confusing than useful. How about something along the lines of "probability 0 is generally understood to represent 'impossibility', while 1 is understood to represent 'certainty'". That makes it clear what the
5064:
4207:
2589:
I am not quite sure as to its relevance here, or whether it should be here at all, but I wonder whether this quote by Niels Bhor would be of use here as an example of probability in action: "I know nothing of probability, I only know that someone once told me that a million monkeys could probably
2045:
is misleading. John von
Neumann wrote an article about "hidden variables" in quantum mechanics in which he proves that a certain quantum mechanical system cannot be described within the usual probability theory framework (measure space, Borel algebra, etc.). The Knowledge article on von Neumann
211:
Perhaps I should stop defending and ask some of my own questions, to wit: what does "impossible" mean, and how can we say with any "certainty" (whatever that means) what is or is not impossible? "Almost everywhere", "almost impossible" and "almost certain" have precise mathematical definitions in
2966:
Two good quality sources comes from
Richard L. Sheaffer in his book, Introduction to Probability and Its Applications, and Glenn Shaffer in his book Probability and Finance. Both Sheaffer and Shaffer are well known mathematicians whose textbooks are used in a vast majority of colleges, including
2190:
That is correct. A xor B is the same event as (A and not B) or (not A and B), in which the two choices for "or" are disjoint events. From that the result follows by applying the given formulas. There are 10 possible combinations involving A and/or B, not counting symmetric combinations twice, of
2123:
Please start new topics at the bottom, not at the top. It's higher because of the "or". It would have been very slightly higher again if the "or" was taken to be an "inclusive or". The example quotes the correct probability if you take "or" to be the "exclusive or". However, the example does not
2118:
I don't contribute on here much, let alone for math, but the equation or at least the example labeled "events that aren't mutually exclusive" is clearly wrong. I'm still learning, but I can tell you that the probability of drawing two specific cards out of a deck is not 103/2704; that's close to
1797:
I see we have a new section "Mathematical treatment", giving a basic treatment that was missing. There is some overlap between this and a later section "Theory", and I think the two should be merged. Also, now "Mathematical treatment" is the first section after the lead section, but usually that
1470:
Would it be worthwhile to extend the physical interpretation of probability to the statistical-mechanics definition of entropy, which suggests that in general, the highest-probability configuration of (insert whatever) is the one generally achieved, and this tends to maximize the disorder in the
165:
The most I will concede is Bremaud's definition (P. Bremaud, An Introduction to Probabilistic Modeling, p.4). The event E such that Prob(E) = 1 is the "almost certain" event, the E' such that P(E') = 0 is the "almost impossible" event. I stress the technical aspect of these definitions: "almost"
1879:
Alright, do events exist? We'll use boolean algebra (only yes's and no's), Yes the exist, ok, now do the PARTIAL-EVENTS exist? i.e. when you have an intent to go to the store, before you have begun moving, does the event exist partially in your mind as a probability? Yes or no. Probability =
319:
By the way, is it only my impression or have people stopped teaching the mathematical definition of probability in stats courses? The only textbook I found which contained a formal definition of probability (as opposed to relying on people's intuition) was pretty old. I base my understanding of
1987:
I have concerns about this opening sentence: 1. The definition seems circular (although one might think that there is no way of breaking out of such a circle when it comes to defining probability). 2. Worse, the definition involves a second-order probability: "CHANCE that something is LIKELY".
1927:
Actually it's not the principle of bi-valence, obviously you mis-understood. There are only only three kinds of truth statements in logic, ALL-Actual-exist-is-true, ALL-Partial-exist-is-true, ALL-Negate-exist-is-true. So no I wasn't speaking about the law of the excluded middle. If something
992:
Now, the Law of Large Numbers; though many texts butcher this deliberately to avoid tedious explanations to the average freshman, the law of large numbers is not the limit stated in this entry. Upon reflection, one can even see that the limit stated isn't well-defined. Fortunately, you present
507:
Now, the Law of Large Numbers; though many texts butcher this deliberately to avoid tedious explanations to the average freshman, the law of large numbers is not the limit stated in this entry. Upon reflection, one can even see that the limit stated isn't well-defined. Fortunately, you present
154:
Of course, but I'm not writing for mathematicians. To ordinary human beings, converting from one notation to another is indeed a "computation", although typing "1÷2=" into a calculator is a pretty simple one. I think it's important to show laymen the different ways they might see probabilities
2931:
Besides the grammar and usage problems, this definition endorses a particular view of the (disputed) definition of probability. Even if it were cleaned up to standard grammar, I don't think an unqualified assertion of one perspective on a disputed topic as fact belongs in the first paragraph.
1641:
By the way, Bayesianism is an approach to scientific reasoning based on the idea that a well-confirmed theory is one that presently has high subjective probability. The name comes from the fact that Bayes's theorem is a central tool in the calculation of the probability of a theory given the
1455:
While I am by no means an expert, I have read the book by Von Neumann and Morgenstern, and I have a hard time understanding how game theory is "strictly based on probability." I'd even go so far as to say that that statement is complete rubbish - game theory may apply probability in solution
1440:
In the section "Formalizing probability" a list of three statements is given, and it is implied that the listed statements are the "laws of probability." While the statements are true, they are not the same as the Kolmogorov axioms that the are linked to in the paragraph. I'd say it's fairly
2427:
There is no mention made of the most important interpretation of probability. Assuming Laplacian Determinism is correct, which most if not all of science assumes, then probability is really a study of observers and their inaccuracies, not of the world they are observing. Purely mathematical
74:
The first paragraph is pure poetry. Some of the second paragraph is not accurate. Probability 0 is not the same as impossible, and probability 1 is not the same as certainty. I fixed the page, then somehow, all the changes got lost. grrr don't have the heart to go back through it right now.
2250:
In the above named section, it says "Frequentists talk about probabilities only when dealing with well defined random experiments." It seems doubtfull if frequentists only deal with experiments: surely any repeatable random process will do, even a natural one. So I am picking on the word
5867:
This "Terminology" section seems specific to the Objectivist camp. Maybe it should be moved to after the "Interpretations" section and renamed "Terminology of the Objectivist camp"? I'm not suggesting that that's the ideal solution, just that it seems like a step in the right direction.
465:"Probability in mathematics" -- strike this section and move some pieces of it elsewhere: move coin flipping & law of large numbers to an application section, move limit definition to discussion of frequentism, strike R&G, mention Bayesianism under a section on interpretation.
127:
the fact that very often, "0" is taken to be impossibility and "1" certainty. Anyway, do please make your point clearer in the text! If not you, who? If not now, when? :-) Also, what's this about "slithering away" on a "common sense" argument? What do you mean? :-) --Slitheringly,
103:
Ok, I just reread the main text. I concede that I did not make the point clear, and it could, should, and probably will be redone by someone smarter than me by the time I peruse down the relevant texts and ponder it a bit. If not, I will be back after I think about it some more.
227:
the mathematics of probability to real-world situations is also a philosophical one, and is the same as the question of how to apply scientific findings to real life. That, too, is a subject entirely irrelevant here. Putting links to articles about those philosophical questions
232:
appropriate here, so feel free to do so. But this article is about probability itself as a subject. The mathematics of probability--which, like all mathematics, exist entire independently of any interpretation or application thereof--assigns the number "1" to mean "certainty",
1672:
I think there ought to be a separate article on chance (including propensity). There's plenty to say about it. Currently I think the whole treatment of the different varieties of probability in connected articles is fairly poor, and could do with improvement & a clean up.
190:
that there are academics, from a wide variety of fields, who dispute about the very questions on which you are dogmatic. Moreover, an encyclopedia article called "probability" should do justice to all sides of this disputation. Anyway, I totally agree with your last sentence!
284:
However, these effects only show up if you do "artificial things" such as flipping a coin forever or picking a random real number. For everyday probabilities, which are always discrete probabilities, the notions of zero probability and impossibility are indeed identical.
2893:
probability of getting exactly pi is 1/infinity = 0. However it is not *impossible* to get pi, just very very unlikely. Of course this is very much a theoretical concept, since in pretty much all real world applications you have only a finite number of possible outcomes.
2621:
This is my first time editing or pointing something out, so I apologize if I'm not using the correct etiquette but the union and intersection operators are currently flipped for most of the article rendering the formulas wrong. I don't have time to change it, good luck.
1821:
Agree, I did pinch some of the text from Prob theory section as it explained some things better. As to position, I've moved it below history. I think there is some text from the edits you mention which could helpfully be reinserted, without the article growing too much.
1811:. While these edits were somewhat extreme, we should guard against the other extreme: a more formal mathematical section growing beyond bounds. Instead we should aim at keeping it confined to an elementary level, referring to other articles for more advanced things. --
1902:, I know of no justification for equating probability with "partial existence" or "partial truth", and I am unable to assign a meaning to the sequence of words "truth does exist all of the time", so the above is totally lost on me. But, in any case, this page is for
1547:
degrees of belief in their truth, or to logically justifiable degrees of belief in their truth. Among statisticians and philosophers, many more distinctions are drawn beyond this subjective/objective divide. See the article on interpretations of probability at the
79:
No mention of subjective probability? It's been too long since I've studied this, so I won't change this article, but clearly we need to say something about that. Someone's also clearly got to write about the different philosophical theories of what probability is.
3032:
To start the discussion- it seems like there's two things battling it out in this article 1) certainty+epistemology and 2) probability+mathematics. This article is nearly all about mathematics but that weird first parag was trying to take it in another direction
5451:
Another edit, adding about 2k chars at this position, was made by the same IP to this talk section at 01:41 (still on 7 July) -- and was reverted (as "Mumbojumbo", per edit summ) at 01:44 by a colleague with (at that time) about 6 months of WP reg'n and 2100
1110:
that the coin was a "fair" coin). Furthermore, mathematically speaking, the above limit is not well-defined; the law of large numbers is a little more convoluted and dependent upon already having some definition of probability. The theorem states that, given
625:
that the coin was a "fair" coin). Furthermore, mathematically speaking, the above limit is not well-defined; the law of large numbers is a little more convoluted and dependent upon already having some definition of probability. The theorem states that, given
2171:
I know next to nothing about probability, but I came here looking for a formula for getting the probability of the exclusive or of two sets. Is this something that could be added to the page, as it's always included in similar discussions on boolean logic.
1618:
has been spammed by multiple IP addresses across articles on several European language Wikipedias. Could regular editors of this article take a look at it to see if it adds high quality, unique information to the external links section in keeping with our
351:
This article needs some work. Different topics are mixed together and there are some inaccuracies. The edits listed below (second list) are intended to separate the article into four parts: (1) concept of probability, (2) an overview of formalization (let
2143:
Independence and mutual exclusion are completely different things. If you draw a card from a 52-card deck, there is a 1/13 chance that it is a Jack, and a 1/4 chance that it is a Spade. These events are independent: the chance to draw a Jack of Spades is
271:
Maybe this example is a tiny bit more down to Earth: say you flip a fair coin repeatedly, keep doing it forever. How likely is it that you get Tail all the way through, forever? Pretty unlikely. In fact, the probability can be shown to be 0. But truly
2428:
discussions on probability may be interesting to mathematicians, but mathematicians are a small subset of people who are interested in the broader concept of probability. Is there not room for other views besides strictly mathematical ones? Thanks --
207:
My entire point really is that the current theory of probability confounds common sense in some aspects. The places where this happens ("impossibility" etc) are a result of the construction of the theory. We should embrace this, not gloass over it.
173:
Further complicating everything is the distinction between the continuous and discrete aspects of probability. I notice that Probability is (lately) classified as discrete math, but thats not entirely correct, and is a topic for a different day. :)
6071:
I don't see how you are getting that implication from the image. Each combination is given the same height on the graph, so it illustrates the point that "The probability of any one combination is identical to that of any other combination, 1/36."
1637:
I published a book on physical probability (chance) recently, and thought I would write a section for this article on chance. At present there is just a very brief section on the propensity interpretation. Or should this be a separate article?
1562:
at all. A frequentist is actually unable to technically interpret such uses of the probability concept, even though 'probability' is often used in this way in colloquial speech. Frequentists only assign probabilities to outcomes of well defined
405:"Representation and interpretation of probability values" -- doesn't mention schools of interpretation; has stuff like "To use the probability in math we must perform the division and express it as 0.5." -- bizarre; odds discussion -- good idea.
300:
In mathematics, the notion of only one of those events occuring is sheer nonsense. (Mathematics is incapable of distinguishing between "can" happen and "does" happen so if only one event "does" happen then that's because it has probability 1.)
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It is often easier for people to understand risk when probabilities are expressed as "natural frequencies" rather than as a number in the range of 0-1. See, for example: The advantages of expressing probabilities as natural frequencies. See:
4286:, because of the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards" but should only be counted once.
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Stoppard's scene that so poignantly illustrates the issues involved with using the law of large numbers to interpret probabilities; the exact issue of it not being a guarantee of convergence. I have an edit which I present for discussion:
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Stoppard's scene that so poignantly illustrates the issues involved with using the law of large numbers to interpret probabilities; the exact issue of it not being a guarantee of convergence. I have an edit which I present for discussion:
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489:"Remarks on probability calculations" -- generalize remark about numbers of possible events, etc., to general modeling considerations; expand remark about Bayes' theorem to mention general strategy (integration) and potential problems.
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Is it quite true that subjective probability is the same as Bayesianism? I was under the impression that Bayesianism is rather a mathematical theory which is an (or perhaps the only current) interpretation of subjective probability.
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Please can an expert add some content to this page on the frequentist, propensity and subjectivist (Bayesian) interpretations of probability as this seems pretty crucial but is missing from the article. There is a starting point
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In older versions there were two more extensive sections, one titled "Formalizing probability", the other "Probability in mathematics", but these were replaced by one very brief brief section "Theory" referring the reader to
3592:; however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken, such as, if a red ball was taken, the probability of picking a red ball again would be
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in a uniform distribution is 1. I think the answer to 212.58.233.129's question is that an event of probability 1 does not imply that it is inevitable, and having a probability of 0 doesn't imply that it will never happen.
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By the way, I haven't added to a Knowledge discussion before so sorry if I haven't followed the proper protocols. I just thought I'd mention my concern in the hope that some hard-core Knowledge person would follow it up.
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An individual point in a continuous uniform distribution has a theoretical probability zero because it is one point in an infinite set of points. The number of points in any segment of the real number line is infinite (see
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para 2: that probabilities are numbers (such as zero or one) is adopted as an axiom in order to construct a mathematical theory of probability; the bit about zero == impossible and one == certain seems out of place in para
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I submit that coins should be assumed fair (unbiased) unless otherwise stated. Specifying a fair/unbiased coin with each use of the word, or that we are drawing a "random" card from a "regular" deck obscures the article.
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An analogy would be, the probability of an individual point in a continuous uniform distribution would be the probability of a winning lottery ticket, in a lottery with an infinite number of tickets (vanishingly small)!
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Nonetheless, I agree that the way probability is `defined' in the current version of the article needs some refinement, and other than the above comment, I like what you've got, Tlee. I'd say just go ahead and make the
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Nonetheless, I agree that the way probability is `defined' in the current version of the article needs some refinement, and other than the above comment, I like what you've got, Tlee. I'd say just go ahead and make the
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I find several of the examples on this page and some the pages linked to it to be confusing. The examples need to be linked to easily understandable, real world situations rather than abstract mathematic explanations.
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Note: Most mathematicians would claim that 1/2 = 0.5 and that no computation is needed to convert them. Likewise, 50% is merely notational abbreviation for 50/100 and likewise needs no conversion to be a real number --
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Even if there are contexts in which probability theory is useful with other definitions of what 0 and 1 mean, that doesn't change their definition in the far more common contexts our readers are likely to care about.
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theory, but I rather suspect none of these academicians are practicing, contemporary mathematicians (unless they be constructivists... and even they would likely uphold the discrete aspect. But lets not go there :).
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it is not. It's just not going to happen. And if you don't like zeros all the way through: how likely is it that you flip the binary expansion of π (say Tail = 0, Head = 1)? Again, the probability is 0. In fact,
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to further our knowledge of probability, we must truly understand the definition of the word first. According to Knowledge, there is not a clear, valid definition for probability, and I propose to change this.
2091:; as editors of an encyclopedia we must instead report what recognized experts have to say on the topic. The issues you raise concern the interpretation of probability and should be dealt with in the article
119:. You might deny that there is such a thing as subjective probability (that it is properly called a type of probability), but that's another kettle of fish; if you deny it in the article, you do not take the
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I have removed the word "likelihood" from the definition of probability because while in common use the two are synonyms, likelihood has a technical meaning in statistics that is distinct from probability.
484:"Distributions" -- restate definition -- distribution as model; state connection to formal theory -- cdf == measure, pdf == basis for constructing cdf; mention Gaussian, t, chi-square, gamma distributions.
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handle it mostly), (3) interpretations of formal theory, and (4) applications. If there is interest, I'll carry out the edits on a trial version so that you can see what the effect of such edits would be.
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has an episode which may be about this subject (if not moving this note to the appropriate talk page earns cookies). You can add it to "External links" by pasting * {{In Our Time|Probability|b00bqf61}}.
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I've often heard "0" probability described as "impossibility" and "1" as "certainty." It is of no explanatory advantage at all to say that "number 0" means "probability 0." The reader already knew
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415:"Remarks on probability calculations" -- "The difficulty of probability calculations lie in..." -- this statement describes one very specific kind of difficulty. para 2, "To learn more..." --
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Someone added it again. "attitude of mind" is not mentioned in the body of the article so it seems pointless, and I have no idea if the reference mentions it. I'm going to delete it again.
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Partial existence, or a partial truth (small truth). Now does truth exist all of the time? Yes or No. Yes. Therefore probability exists, of the type that it is a partially-existent-truth.
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A Bayesian may assign a probability to the proposition that 'there was life on Mars a billion years ago', since that is uncertain, whereas a frequentist would not assign probabilities to
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For the reasons stated above, the "definition" given in this section is wrong and misleading for any serious readers. I would try to rewrite this section soon. Are there any objections?
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For example, the probability that aliens will come and destroy us is x to 1 because there are so many different possibilities (nice aliens, mean but weaker aliens, non existing aliens)/
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In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an
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In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an
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494:"See also" -- add links to gambling and decision theory -- there should probably be an article devoted entirely to gambling calculations, but that's beyond the scope of this article.
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to do with the intuitive notions of probability people use in everyday life. In everyday life, a set of events each have a probability of happening and one of those events happens.
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make the refs' links back into this section have an effect only for those who've already found which section causes them to appear at the bottom, and "deactivated" the "hiding".
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There should be a mention of probability in the luck article, or is there no connection between luck and probability? ISn't luck after all probability resulting in our favour?
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I like the attempt in the article to not just define the mathematical terms, but some philosophical aspect as well. The reference is to Kendall, who for sure is not the least.
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Needs to have basic mathematical definition of probability and rules for p(A and B) p(A or B) p(A|B). The applications section needs some attention to check for POV issues. --
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which a formula is given for four. The formulas for the other six can be derived from those. I do not have a clear argument for or against listing these in the article. --
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has been reduced to a stub, it may be best to redirect it to here, after merging any valuable information left not already present here (possibly none – didn't check). --
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I agree. I've tried to reword it to fix these problems. I also deleted the reference to the old dictionary definition, since it no longer applies to the current text. --
1582:, in which situations are given integer values greater than 1. This is an extension of the multi-dimensional space intrinsic to M-theory and modern theoretical physics.
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it seems these types of statements continue to creep in the article. There are plenty of modern sources to cite that establish the meaning of probabilities of 0 and 1.
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This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class.
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so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play. Flipping a coin is 50-50. Throwing a
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Sheaffer, R. (2010). Introduction to probability and its applications. (3rd ed.). Glenn Shaffer. (2010, June 18). Probability and Finance. Retrieved June 22, 2012 from
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this is true. Please can someone explain this. I have basic probability knowledge and would like to understand more into depth the theories and ideas of probabilities.
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Open to discussion if anyone really feels strongly that all of this stuff needs to be in a non-technical summary article, but I feel pretty strongly that it does not.
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The section "History" contains the following mysterious statement: "He deduced a formula for the mean of three observations." I assume that this is not the formula (x
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As a question, would it be better to use the term independent rather than mutually exclusive? Furthermore, can you explain what is "inclusive" versus "exclusive or?
1501:, because it seems like it would fit a lot better into this article. However, I don't understand this text well enough to merge it. Can anyone help me merge it in?
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Yes calling probability an "attitude of mind" is extremely confusing and wrong. I've changed it once but it was reverted without good reason ("not an improvement")
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article do most of the work; mention density & distribution functions and connect those with Kolmogorov formulation; mention Cox formulation and give cross ref.
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Brown, T A, Shuford, E., "Quantifying Uncertainty Into Numerical Probabilities for the Reporting of Intelligence," Rand Corp., Santa Monica CA, R-1185-ARPA, '73.
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Whatever point the image is trying to make, it requires A) movement to another part of the page or another page altogether, B) clearer labelling, or C) removal
4874:{\displaystyle {\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}}
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refers to the fact that these events both occur (as does any other event) with probability 0, that is, with no measure. This stuff is really sticky, no doubt.
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For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is
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i.e. we can think of a picture that fades into black, and then back to it's original state, as it fades into black, its 'likelyness' of existence decreasing?
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Oh, grow up. I am expressing my opinion about what would be useful here; if you disagree, express yours. If you think I'm full of crap, say so, but explain
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Lee, you are speaking for all of our readers? Well, then, by all means I bow to your authority. My participation in this conversation is necessarily over.
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5059:{\displaystyle {\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}}
4202:{\displaystyle \mathrm {P} \left(A{\hbox{ or }}B\right)=\mathrm {P} \left(A\right)+\mathrm {P} \left(B\right)-\mathrm {P} \left(A{\mbox{ and }}B\right).}
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410:"Distributions" -- definition is incorrect; doesn't state any connection to probability formalization; doesn't mention important continuous distributions
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479:"Representation and interpretation of probability values" -- mention schools of interpretation (Bayesian, frequentist) here; strike "1/2 == 0.5" stuff.
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as such devoid of meaning: you just prove what the axioms allow you to prove. But of course the whole point of the axiomatic system is that there are
316:. The only kind of interpretation of QM compatible with mathematical probability is the one given by Many-Worlds. Again, this is not a trivial issue.
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Lest this not be construed as a neutral point of view, open any text on the foundations of probability and measure theory. That's where I learned it.
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It's clear that the article is starting with possible results on a single six-sided die. So please do not "correct" it to "a dice" or "dice roll."
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There's no winning on this issue. Likelihood is circular too. Removing "chance that something is likely" is probably (er, likely?) a good thing.
2876:. The converses of these statements would be that all events of probability 0 are impossible, and that all events of probability 1 are certain.
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Situations do arise where probability theory is somewhat lacking. One method of attempting to circumvent this indeterminancy is the theory of
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There are many interpretations that are made of probability, but I find the opening sentences of this article to be confusing and incomplete.
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Hello Everyone, I was just wondering if someone could create a sub-page of how probability has been used culturally and in fiction, thanks.
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is the article about the mathematical theory; the details and axioms and theorems are explored in a number of even more specific articles (
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This is a misleading image: it implies that there are different probabilities for different combinations of fair dice, but there aren't.
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It will be helpful to include a discussion of natural frequencies and the advantages cited in these references in this article. Thanks! --
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where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as
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probability is and how probability claims are to be interpreted, as though this were "known by scientists," you fail to do justice to the
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2697:"An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true"
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1. Experience: relevant past information. 2. Counting: abstract reasoning. 3. Subjective: instinctive feelings, knowledge, training.
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I suggest that the discussion above, which appears to be inactive, be archived. I'll do so in a week or two unless I hear otherwise.
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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Epstein, E. S., "A Scoring System for Probability Forecasts of Ranked Categories," J. Appl. Meteorology, 8 (6), 12/69, pp. 985-987.
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para 1: "probability theory" doesn't, and can't, say anything about what words mean; "probability theory" is out of place in para 1.
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on the interval , every point has the same probability, which must be 0. Do you think this means that every point is impossible? --
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The textbook "Essentials of finite mathematics," Robert F. Brown, Brenda W. Brown has these three distinctive views of this term:
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2691:"If probability is equal to 1 then that event is certain to happen and if the probability is 0 then that event will never occur."
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I found the frequentist/Bayesian distinction unclear and misleading. I believe Brown/Brown got it right with these 3 categories.
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Come on, that was the only sentence in the whole article that I understood! The rest of it is all dictionary-y and in Latin! --
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Landa, S.,"CAAPM: Computer Aided Admissible Probability Measurement on Plato IV," Rand Corp., Santa Monica CA, R-1721-ARPA, '76
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923:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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Well, at least one reader now knows he doesn't know anything about probability, a nice improvement over merely thinking it. ;-)
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5612:, made Improbable a redirect to Probability, because.. of course. And then a bot fixed the double redirect, and there you go,
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This is a very nice article. I was very happy to see the distinction between 0 probability events and impossible events. :) --
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The immediately following {{refs}} invocation has been added by me, since neglect of this measure has been causing the refs to
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I see it has already been reverted. I don't think this is a non-technical article. It should have at least high school math.
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4574:(Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).
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imply, for example, that if you checked the facts an infinite number of times, you'd get two answers with equal frequency. --
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will get closer and closer to the probability of a single coin flip being heads. Most casual observers are even willing to
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will get closer and closer to the probability of a single coin flip being heads. Most casual observers are even willing to
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para 2: strike last sentence and move discussion of numerical values of "impossible" and "certain" to a math-oriented part.
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by this expression. However, it is possible to define a conditional probability for some zero-probability events using a
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1918:. This page is not intended for a general discussion on the concept of probability. Thank you for your understanding. --
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In everyday life, it takes many iterations of an experiment, each with its singular outcome, to be able to reconstruct a
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appear, without explanation after (well, or within) the section that is, at any given time, temporarily the last section
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para 2: what mathematicians think probability is, is not too important. Disclaimer: I have an undergraduate math degree.
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since we're talking reals). When dealing with infinitesimals, it may not be appropriate to define it that way. And it
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For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is
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have in this article actually, that wouldn't fit better under some other heading. Maybe the best thing to do with
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470:"Formalization" -- strike para 1; rephrase existing discussion as brief overview of Kolgorov formulation and let
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para 1: probable == "likely to..." is problematic -- what is "likely" ? This will lead to a circular definition.
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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from one in a million. I'm not sure what the author was trying to describe by this section, but it didn't work.
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Gao, J.Z.; Fong, D.; Liu, X. (April 2011). "Mathematical analyses of casino rebate systems for VIP gambling".
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Actually, I dislike all my tries as soon as I write them down. But there will be a new version real soon now.
958:, so the edit history is now with the archive page. I've copied back the most recent thread. Hope this helps,
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Savage, L J, "Elicitation of Personal Probabilities & Expectations," J. Amer. Stat. Assoc. '71, 783-801.
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about the probability. In mathematics, all of the outcomes happen and the probability distribution is exact.
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relate to the formula that it purports to represent with respect to mutual exclusivity, so I deleted it. --
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concepts, but it isn't "based on probability." This is even obvious from the article on game theory itself.
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This aspect of the law of large numbers is sometimes troubling when applied to real world situations. ...
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This aspect of the law of large numbers is sometimes troubling when applied to real world situations. ...
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Shuford, E., Brown, T.A., "Elicitation of Personal Probabilities," Instructional Sci. 4, '75, pp. 137-188
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Eisenberg, E., Gale, G. "Consensus of Subjective Probabilities," Ann. Math. Stat., 30, 3/59, pp. 165-168.
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Perhaps you are not understanding the point of my comments, so let me try to be clearer. When speaking of
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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Winkler, R. L., "Scoring Rules and Probability Assessors," J. Amer. Stat. Assoc., 64, '69, pp. 1073-78.
5694:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit
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write the whole of my oeuvre on a typewriter in a million billion years, so I punched him in the nose."
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Shuford, E, Albert, A., Massengill, H., "Admissible Probability," Psychometrika, 31, '66, pp. 125-145.
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References could include such items as the following and SHOULD INCLUDE Savage, Eisenberg/Gale items:
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or both events occur on a single performance of an experiment this is called the union of the events
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1771:"between 0 (no chance of occurring) and 1 (will always occur)" :- Shouldn't this line be changed to '
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https://web.archive.org/web/20160203070724/http://statprob.com/encyclopedia/AdrienMarieLegendre.html
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http://www.bmj.com/rapid-response/2011/11/02/advantage-expressing-probabilities-natural-frequencies
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Brown, T. A., Shuford, E., Quantifying Uncertainty Into Numerical Probabilities, R-1185-ARPA, '73.
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I think we should do nearly the opposite thing: move most of the contents in this article to the
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The question of what "certainty" means is a philosophical one, indeed it is the whole subject of
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before doing mass systematic removals. This message is updated dynamically through the template
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Probability is a way of expressing knowledge or belief that an event will occur or has occurred.
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4075:{\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.}
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it is the prediction of an event happening based on an already occurrence of that same event
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The first sentence is not correct (the second one is). Perhaps some clarification is needed.
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Good, I J, "Rational Decisions," J. Royal Statistical Society, Ser. B, 14, '52, pp. 107-114.
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as defined above in the theory section. For another illustration of the differences see the
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The probability of any one combination is identical to that of any other combination, 1/36.
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Brown, T. A., Probabilistic Forecasts and Reproducing Scoring Systems, RM-6299-ARPA, 6/70.
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McCarthy, J., "Measurement of Value of Information," Proc. Nat. Acad Sci., '56, pp. 654-5.
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para 3: appears to be a link collection; also contains a misrepresentation of "statistics".
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sequence you produce this way has probability zero, even though one of them will happen.
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is simply the ratio of the probabilities of the two events. When arbitrarily many events
3465:{\displaystyle \mathrm {P} (A\mid B)={\frac {\mathrm {P} (A\cap B)}{\mathrm {P} (B)}}.\,}
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probably get close to the stated probability, and provides no guarantees of convergence.
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probably get close to the stated probability, and provides no guarantees of convergence.
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That depends on whether your definition of "=" implies that 0 = 1/∞ (or more exactly, 1/ℵ
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2148:. They are obviously not mutually exclusive. For "inclusive" versus "exclusive" or, see
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2042:"This means that "deep inside" nature can only be described using probability theory."
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redirects here, but im gonna steal it for an epistemology acrticle. will do a disambig
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If you found an error with any archives or the URLs themselves, you can fix them with
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4279:{\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}}}
1366:{\displaystyle \Pr \left(\lim _{N\rightarrow \infty }{N_{H} \over N}=\Pr(H)\right)=1.}
881:{\displaystyle \Pr \left(\lim _{N\rightarrow \infty }{N_{H} \over N}=\Pr(H)\right)=1.}
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2046:(quantum mechanics section) says this paper contained a conceptual error, however.
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1115:) and any arbitrarily small probability ε and difference δ, there exists some number
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Last edited at 14:38, 9 September 2007 (UTC). Substituted at 15:35, 1 May 2016 (UTC)
630:) and any arbitrarily small probability ε and difference δ, there exists some number
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SpatialOS redirects here. Why? There is no mention on this article of SpatialOS.
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occur on a single performance of an experiment, this is called the intersection or
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5198:{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,}
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article; doesn't mention Cox formulation as alternative to Kolmogorov formulation.
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Brown, T. A., An Experiment in Probabilistic Forecasting, R-944-ARPA, July '73.
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point of view on the issue; you even signed them. That is against our policy of
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If you wish to start a new discussion or revive an old one, please do so on the
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Could someone here help arbitrate on a discussion regarding probability on the
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Quotations don't have citations of the works in which they originally appeared.
335:
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5726:. No special action is required regarding these talk page notices, other than
1942:
I make a claim that this page is not intended for this kind of discussion. --
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people have transported into physics without justification; an intuition with
6046:
Graphic titled 'The probabilities of rolling several numbers using two dice.'
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would consider some events with exactly zero probability as possible anyway.
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para 3: strike this para; put link collection at end of introductory section.
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Theory of the Combination of Observations Least Subject to Errors: Part One
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Probability is the chance that something is likely to happen or be the case
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probability. Attributing any other meaning is philosophy, not mathematics.
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is the set of physical phenomena associated with the presence and flow of
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2095:. If you cannot find your point of view represented there, find published
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and is supposed to express all other particles also the "theory of all"
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for example, if two coins are flipped the chance of both being heads is
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para 2: strike sentence 1 and modify remainder of paragraph accordingly.
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3193:. As an example, the chance of not rolling a six on a six-sided die is
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https://rogerkerry.wordpress.com/2016/09/30/communicating-risk-part-2/
3793:{\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.}
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para 1: strike mention of "probability theory" -- there are links to
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I've removed this seemingly nonsensical statement from the article:
2251:"experiment", here and in the next bits, as being far too specific.
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that present that point of view, and report on that, while properly
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Could it also be: Partial-ability? or Partially existent occurance?
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I'd like the article to emphasize that mathematical probability has
5963:(die) is a 1-out-of-6 probability. Picking a card put of a deck of
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is the probability that it will occur, given that some other event
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What is lacking here, I think, is a good common effort to turn the
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para 1: describe "probability" in terms of expectation or betting.
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The formula is just P(A xor B) = P(A) + P(B) - 2P(A and B) right?
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also occurs or has occurred. Conditional probability is written
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1211:{\displaystyle Pr\left(\left|\Pr(H)-{N_{H} \over N}\right|: -->
726:{\displaystyle Pr\left(\left|\Pr(H)-{N_{H} \over N}\right|: -->
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just a matter of definition, as is all of mathematics. --LDC
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since only 1 red and 2 blue balls would have been remaining.
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I've removed the following sentence fragment from the lede:
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No mention of them, I would like to see an example of them.--
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Is probability, better understood as PARTIAL EXISTENTIALITY?
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are of interest, not just two, the rule can be rephrased as
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plain and simple is to turn it into an disambiguation page?
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For example, the chance of rolling a 1 or 2 on a six-sided
5919:
all your talk page messages with four tildes (~~~~) — See
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for additional information. I made the following changes:
5593:
I suppose...because SpacialOS is made by a company called
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3728:{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B),\,}
1092:{\displaystyle \Pr(H)=\lim _{N\to \infty }{N_{H} \over N}}
607:{\displaystyle \Pr(H)=\lim _{N\to \infty }{N_{H} \over N}}
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http://statprob.com/encyclopedia/AdrienMarieLegendre.html
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This section is very unclear, and probably (sic) wrong.
3951:{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B).}
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I just want to point out that if you dogmatically state
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So the main page was changed back. Fine. wiki wiki wiki
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If I am getting this wrong can you please explain why.
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probability to a particular outcome (in this case, our
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probability to a particular outcome (in this case, our
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252:. A thick skin is a useful tool for this place. --LDC
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links to the more thorough, techincal articles (e.g.
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I have replaced the word "experiment" by "trial". --
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1/27. Why would the probability be higher than 1/52?
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I'd love if someone can clear this up for me thanks.
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Talk:Frequency probability#What kind of experiments?
396:"Formalization" -- section recapitulates content of
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surely "1" could be interpreted as "certainty." See
5730:using the archive tool instructions below. Editors
5600:Okay, it's actually not a joke. Experienced editor
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posterior is proportional to prior times likelihood
2955:User:Nparibello-NJITWILL/Summer 2012 | Probability
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3242:{\displaystyle =1-{\tfrac {1}{6}}={\tfrac {5}{6}}}
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2803:{\displaystyle \int _{0}^{1}f(x)=1\implies f(x)=1}
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1838:Question on the word and definition of probability
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419:article doesn't have any computational stuff, and
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223:, and is irrelevant here. The question of how to
3018:I've changed it again. Let's hope for discussion
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2617:problem with the Mathematical Treatment Section
1866:I cannot say in truth that this contributes to
919:, and are posted here for posterity. Following
5716:This message was posted before February 2018.
4089:If the events are not mutually exclusive then
2688:One of the first sentences on the article is:
1798:position is reserved for the History section.
975:talk page if you have any questions. Thanks
4360:, before (prior to) and after (posterior to)
3557:of such events (such as those arising from a
1969:Talk:Probability theory#Central Limit Theorem
1914:, which means it must have been published in
1612:
1527:, that are outcomes of actual or theoretical
913:The comment(s) below were originally left at
3862:then the probability of either occurring is
3181:not occurring); its probability is given by
1971:, which is where this appears to belong. --
1779:always occur' with a link to the article on
5638:, which I should have thought to check. —
5369:is the "only fundamental" proposed particle
2704:They are both true are they not? Both say:
138:I'm going to lose, with absolute certainty.
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5686:I have just modified one external link on
5359:fudamental particle / theory of everything
5967:is 1-out-of-52 (no jokers) probability.
4527:{\displaystyle P(A|B)\propto P(A)P(B|A)}
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1615:. giacomo.lorenzoni.name/arganprobstat/
989:First, good job on the entry to all....
504:First, good job on the entry to all....
432:Proposed edits to address issues above:
178:class in this will set us all straight.
5931:. Other contributors might disagree. -
5863:"Terminology of the probability theory"
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2447:, which is a better place for this. --
1894:I do not particularly subscribe to the
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5969:2601:582:C480:BCD0:BC4A:2142:18E9:523F
44:Do not edit the contents of this page.
3542:{\displaystyle \mathrm {P} (A\mid B)}
3367:{\displaystyle \mathrm {P} (A\mid B)}
3177:is the event (that is, the event of
2973:http://www.probabilityandfinance.com/
2114:Events That Aren't Mutually Exclusive
954:I've moved the existing talk page to
5798:and Communicating Risk: Part 2 See:
5597:. Ha ha ha. 17:49, 5 July 2017 (UTC)
1567:, that is, where there is a defined
1279:). That is, convergence will occur
1262:should guarantee the convergence of
1218:\delta \right)<\epsilon }" /: -->
794:). That is, convergence will occur
777:should guarantee the convergence of
733:\delta \right)<\epsilon }" /: -->
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1550:Stanford Encyclopedia of Philosophy
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5836:Both are correct as singular. See
5604:created the page as a redirect to
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3374:, and is read "the probability of
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3135:Removing massive technical section
1655:Hi Richard - go for it! Remember,
1515:will assign probabilities only to
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1134:\delta \right)<\epsilon }": -->
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649:\delta \right)<\epsilon }": -->
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5690:. Please take a moment to review
4863:if A and B are mutually exclusive
4705:{\displaystyle P(A^{c})=1-P(A)\,}
3505:{\displaystyle \mathrm {P} (B)=0}
2921:Removal of sentence added to lede
2377:) 09:56, 30 June 2009 (UTC) hi!
1775:no chance of occuring' and 'will
1737:That sounds really good to me...
921:several discussions in past years
5817:Die is singular, dice is plural.
1686:Merge Probability theory to here
29:
3253:for a more complete treatment.
2735:It's been a while, but I think
2707:If P(A) = 0 that is impossible
2654:Do you hav' a source for that?
1633:New Section on Physical Chance?
423:is pretty light on computation.
5996:International Gambling Studies
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5955:Probability is used to design
5941:08:27, 30 September 2021 (UTC)
5908:03:35, 30 September 2021 (UTC)
5878:22:55, 28 September 2021 (UTC)
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2443:We have a separate article on
1793:Mathematical treatment section
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3156:. Here is the text removed:
3091:22:39, 14 December 2014 (UTC)
2898:23:37, 23 February 2007 (UTC)
2820:03:09, 24 February 2007 (UTC)
2810:, so the probability of each
2710:If P(A) = 1 that is certain.
2684:A is certain implies P(A) = 1
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2413:15:32, 23 December 2006 (UTC)
2332:04:34, 11 February 2009 (UTC)
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1947:14:44, 22 November 2007 (UTC)
1938:13:10, 22 November 2007 (UTC)
1923:12:08, 21 November 2007 (UTC)
1890:06:37, 21 November 2007 (UTC)
1875:16:23, 20 November 2007 (UTC)
1861:08:47, 20 November 2007 (UTC)
1832:07:58, 14 November 2007 (UTC)
1816:00:30, 14 November 2007 (UTC)
1788:13:13, 11 November 2007 (UTC)
1762:04:22, 10 November 2007 (UTC)
1506:17:55, 28 February 2007 (UTC)
1413:22:31, 25 February 2006 (UTC)
1403:18:20, 21 February 2006 (UTC)
1254:is just the sample mean of a
939:14:38, 9 September 2007 (UTC)
769:is just the sample mean of a
6008:10.1080/14459795.2011.552575
5857:02:01, 25 October 2018 (UTC)
5831:00:51, 25 October 2018 (UTC)
5563:, :57, 12 August 2015 (UTC)
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2526:05:03, 29 January 2010 (UTC)
2423:18:43, 12 January 2007 (UTC)
2393:03:55, 19 January 2010 (UTC)
1976:05:21, 24 January 2008 (UTC)
1712:page and parts of it to the
1212:\delta \right)<\epsilon }
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727:\delta \right)<\epsilon }
7:
5927:Noted. That's why you made
5463:08:22, 24 August 2015 (UTC)
5242:22:23, 23 August 2015 (UTC)
4554:varies, for fixed or given
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3043:08:02, 16 August 2012 (UTC)
3028:07:55, 16 August 2012 (UTC)
3008:07:51, 16 August 2012 (UTC)
2659:15:25, 4 January 2013 (UTC)
2445:probability interpretations
2199:Mean of three observations.
2093:Probability interpretations
1720:, I think suits best as an
1714:Probability interpretations
1589:Probability interpretations
1476:05:25, 19 August 2005 (UTC)
1461:14:02, 2 January 2007 (UTC)
1451:Game theory and probability
1446:13:50, 2 January 2007 (UTC)
1260:strong law of large numbers
1023:) of flipping heads as the
1000:gets larger and larger, we
775:strong law of large numbers
538:) of flipping heads as the
515:gets larger and larger, we
10:
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6082:12:37, 1 August 2023 (UTC)
5784:14:58, 2 August 2017 (UTC)
5747:(last update: 5 June 2024)
5683:Hello fellow Wikipedians,
5355:probabilon - the proposed
5048:if A and B are independent
4640:{\displaystyle P(A)\in \,}
3851:{\displaystyle P(A\cup B)}
3559:continuous random variable
3318:
3304:{\displaystyle P(A\cap B)}
2862:02:19, 22 April 2013 (UTC)
2847:02:13, 22 April 2013 (UTC)
2676:04:46, 4 August 2013 (UTC)
2399:Three types of probability
2314:04:32, 11 March 2007 (UTC)
2161:21:34, 25 April 2008 (UTC)
2134:11:19, 20 April 2008 (UTC)
1742:16:40, 21 March 2007 (UTC)
1729:16:26, 21 March 2007 (UTC)
1703:10:20, 21 March 2007 (UTC)
1678:10:09, 20 March 2007 (UTC)
1628:14:54, 14 March 2007 (UTC)
1431:08:01, 26 March 2006 (UTC)
1394:Probability in mathematics
6066:23:40, 31 July 2023 (UTC)
4583:Summary of probabilities
2990:22:01, 26 June 2012 (UTC)
2077:"chances are zero or 100%
2057:04:51, 21 June 2007 (UTC)
2029:03:06, 20 June 2007 (UTC)
2016:02:41, 20 June 2007 (UTC)
1664:07:58, 7 March 2007 (UTC)
1649:23:12, 6 March 2007 (UTC)
1601:16:42, 7 March 2007 (UTC)
1487:17:24, 1 March 2007 (UTC)
1256:Bernoulli random variable
1234:03:44, 13 Apr 2004 (UTC)
962:04:44, 10 Aug 2004 (UTC)
956:Talk:Probability/Archive1
928:
916:Talk:Probability/Comments
771:Bernoulli random variable
749:03:44, 13 Apr 2004 (UTC)
338:, Wednesday, May 22, 2002
5977:14:24, 4 June 2022 (UTC)
5669:08:24, 7 July 2017 (UTC)
5652:18:10, 5 July 2017 (UTC)
5630:18:01, 5 July 2017 (UTC)
5588:15:44, 5 July 2017 (UTC)
5491:21:47, 5 Jun 2005 (UTC)
5443:) 00:50, 12 August 2015
5403:) 00:50, 12 August 2015
5274:Olofsson (2005) page 35.
5265:Olofsson (2005) page 29.
4578:Summary of probabilities
2915:06:15, 2 July 2014 (UTC)
2886:12:39, 3 June 2014 (UTC)
2359:15:35, 10 May 2009 (UTC)
2286:13:30, 22 May 2008 (UTC)
2271:09:47, 19 May 2008 (UTC)
2261:16:44, 14 May 2008 (UTC)
2240:11:58, 12 May 2008 (UTC)
1753:WikiProject class rating
1535:assign probabilities to
1389:23:47, 15 Apr 2004 (UTC)
904:23:47, 15 Apr 2004 (UTC)
361:13:48, 3 Jan 2004 (UTC)
347:13:48, 3 Jan 2004 (UTC)
5679:External links modified
5543:probabilistic potential
5471:Accuracy of Probability
5256:Olofsson (2005), page 9
3624:Independent probability
3327:conditional probability
3321:Conditional probability
3315:Conditional probability
2185:00:07, 9 May 2008 (UTC)
1904:discussing improvements
123:. Moreover, you should
114:subjective probability,
5515:06:06, 10 April 2013
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4085:Not mutually exclusive
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1900:Principle of bivalence
1896:Law of excluded middle
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4355:
4353:{\displaystyle A_{2}}
4328:
4326:{\displaystyle A_{1}}
4281:
4204:
4077:
3953:
3853:
3795:
3730:
3615:
3587:
3544:
3507:
3467:
3369:
3306:
3244:
2805:
2298:
1994:comment was added by
1961:Central Limit Theorem
1767:Question on Statement
1607:Spammed external link
1573:two envelopes problem
1368:
1214:
1094:
883:
729:
609:
314:no mathematical basis
121:neutral point of view
42:of past discussions.
5947:Gambling Probability
5728:regular verification
5658:Gamall Wednesday Ida
5640:Gamall Wednesday Ida
5636:Improbable (company)
5618:Gamall Wednesday Ida
5076:
4891:
4722:
4657:
4602:
4558:
4538:
4466:
4442:
4415:
4388:
4368:
4337:
4310:
4216:
4096:
3969:
3869:
3858:. If two events are
3827:
3742:
3647:
3596:
3568:
3516:
3479:
3389:
3382:". It is defined by
3341:
3280:
3200:
2854:Jim.Callahan,Orlando
2839:Jim.Callahan,Orlando
2739:
2722:uniform distribution
2126:David from Downunder
2089:no original research
1543:according either to
1293:
1135:
1038:
985:Law of Large Numbers
808:
650:
553:
500:Law of Large Numbers
5789:Natural Frequencies
5718:After February 2018
4584:
4290:Inverse probability
3613:{\displaystyle 1/3}
3585:{\displaystyle 1/2}
3251:Complementary event
2756:
2664:Likelihood function
2083:mini-essays giving
1722:disambiguation page
1436:Laws of Probability
1019:the probability Pr(
534:the probability Pr(
5772:InternetArchiveBot
5723:InternetArchiveBot
5327:Probability axioms
5323:Probability Theory
5195:
5194:
5056:
5054:
5050:
5044:
4871:
4869:
4865:
4859:
4702:
4701:
4637:
4636:
4582:
4564:
4544:
4524:
4448:
4428:
4401:
4374:
4350:
4323:
4298:and applications,
4296:probability theory
4276:
4274:
4259:
4244:
4229:
4199:
4186:
4117:
4072:
4067:
4052:
4037:
3986:
3948:
3886:
3860:mutually exclusive
3848:
3803:Mutually exclusive
3790:
3785:
3770:
3755:
3725:
3724:
3664:
3610:
3582:
3539:
3502:
3462:
3461:
3364:
3301:
3239:
3237:
3222:
3150:Probability axioms
3142:Probability theory
2800:
2781:
2776:
2742:
2565:Dubious statement?
2537:The BBC programme
2365:Probability trees?
2338:Confusing examples
1930:BeExcellent2every1
1882:BeExcellent2every1
1870:understanding. --
1853:BeExcellent2every1
1804:Probability theory
1710:Probability theory
1692:Probability theory
1565:random experiments
1499:probability theory
1363:
1319:
1208:
1119:such that for all
1089:
1071:
1025:mathematical limit
972:Answers in Genesis
960:Wile E. Heresiarch
909:Assessment comment
878:
834:
723:
634:such that for all
604:
586:
540:mathematical limit
472:probability theory
443:probability theory
417:probability axioms
398:probability theory
359:Wile E. Heresiarch
354:probability theory
345:Wile E. Heresiarch
295:absolutely nothing
5924:
5894:comment added by
5748:
5634:And now there is
5564:
5562:
5559:) 00:50, :54, :55
5551:comment added by
5521:comment added by
5444:
5435:comment added by
5428:
5426:
5415:comment added by
5404:
5395:comment added by
5388:
5386:
5375:comment added by
5331:Probability space
5208:
5207:
5191:
5136:
5049:
4864:
4567:{\displaystyle B}
4547:{\displaystyle A}
4451:{\displaystyle A}
4377:{\displaystyle B}
4364:on another event
4273:
4258:
4243:
4228:
4185:
4116:
4066:
4051:
4036:
3985:
3885:
3784:
3769:
3754:
3663:
3455:
3266:joint probability
3236:
3221:
3146:Probability space
2980:comment added by
2716:--212.58.233.129
2613:
2604:comment added by
2529:
2512:comment added by
2383:comment added by
2349:comment added by
2146:1/13 × 1/4 = 1/52
2007:
1716:page. This page,
1700:
1580:super-probability
1335:
1304:
1283:, or equivalently
1184:
1087:
1056:
944:
943:
850:
819:
798:, or equivalently
699:
602:
571:
67:
66:
54:
53:
48:current talk page
6089:
6019:
6018:
5991:
5914:
5910:
5782:
5773:
5746:
5745:
5724:
5560:
5546:
5530:
5430:
5424:
5410:
5390:
5384:
5370:
5339:Bryanrutherford0
5291:Bryanrutherford0
5275:
5272:
5266:
5263:
5257:
5254:
5204:
5202:
5201:
5196:
5192:
5190:
5176:
5157:
5142:
5137:
5135:
5121:
5101:
5065:
5063:
5062:
5057:
5055:
5051:
5047:
4971:
4936:
4880:
4878:
4877:
4872:
4870:
4866:
4862:
4711:
4709:
4708:
4703:
4675:
4674:
4646:
4644:
4643:
4638:
4585:
4581:
4573:
4571:
4570:
4565:
4553:
4551:
4550:
4545:
4533:
4531:
4530:
4525:
4517:
4482:
4457:
4455:
4454:
4449:
4437:
4435:
4434:
4429:
4427:
4426:
4410:
4408:
4407:
4402:
4400:
4399:
4383:
4381:
4380:
4375:
4359:
4357:
4356:
4351:
4349:
4348:
4332:
4330:
4329:
4324:
4322:
4321:
4285:
4283:
4282:
4277:
4275:
4266:
4260:
4251:
4245:
4236:
4230:
4221:
4208:
4206:
4205:
4200:
4195:
4191:
4187:
4183:
4172:
4164:
4153:
4145:
4134:
4126:
4122:
4118:
4114:
4103:
4081:
4079:
4078:
4073:
4068:
4059:
4053:
4044:
4038:
4029:
3987:
3983:
3957:
3955:
3954:
3949:
3887:
3883:
3857:
3855:
3854:
3849:
3807:If either event
3799:
3797:
3796:
3791:
3786:
3777:
3771:
3762:
3756:
3747:
3734:
3732:
3731:
3726:
3665:
3661:
3619:
3617:
3616:
3611:
3606:
3591:
3589:
3588:
3583:
3578:
3548:
3546:
3545:
3540:
3523:
3511:
3509:
3508:
3503:
3486:
3471:
3469:
3468:
3463:
3456:
3454:
3444:
3438:
3422:
3416:
3396:
3373:
3371:
3370:
3365:
3348:
3310:
3308:
3307:
3302:
3248:
3246:
3245:
3240:
3238:
3229:
3223:
3214:
3196:
3192:
3160:the text removed
2992:
2934:Bryanrutherford0
2874:Converse (logic)
2809:
2807:
2806:
2801:
2755:
2750:
2599:
2528:
2506:
2395:
2361:
2147:
2097:reliable sources
1989:
1916:reliable sources
1699:
1525:random variables
1372:
1370:
1369:
1364:
1356:
1352:
1336:
1331:
1330:
1321:
1318:
1275:to the mean, Pr(
1219:
1216:
1215:
1209:
1201:
1197:
1190:
1186:
1185:
1180:
1179:
1170:
1098:
1096:
1095:
1090:
1088:
1083:
1082:
1073:
1070:
966:Request for help
926:
925:
918:
887:
885:
884:
879:
871:
867:
851:
846:
845:
836:
833:
790:to the mean, Pr(
734:
731:
730:
724:
716:
712:
705:
701:
700:
695:
694:
685:
613:
611:
610:
605:
603:
598:
597:
588:
585:
63:
56:
55:
33:
32:
26:
18:Talk:Probability
6097:
6096:
6092:
6091:
6090:
6088:
6087:
6086:
6048:
6029:
6024:
6023:
6022:
5992:
5988:
5957:games of chance
5949:
5889:
5885:
5865:
5819:
5791:
5776:
5771:
5739:
5732:have permission
5722:
5696:this simple FaQ
5681:
5576:
5536:
5523:110.175.213.190
5516:
5509:
5497:
5485:
5473:
5361:
5287:
5282:
5281:
5279:
5278:
5273:
5269:
5264:
5260:
5255:
5251:
5213:
5177:
5153:
5143:
5141:
5122:
5102:
5100:
5077:
5074:
5073:
5053:
5052:
5045:
5012:
4991:
4990:
4967:
4932:
4916:
4894:
4892:
4889:
4888:
4868:
4867:
4860:
4824:
4803:
4802:
4747:
4725:
4723:
4720:
4719:
4670:
4666:
4658:
4655:
4654:
4603:
4600:
4599:
4580:
4559:
4556:
4555:
4539:
4536:
4535:
4513:
4478:
4467:
4464:
4463:
4443:
4440:
4439:
4422:
4418:
4416:
4413:
4412:
4395:
4391:
4389:
4386:
4385:
4369:
4366:
4365:
4344:
4340:
4338:
4335:
4334:
4317:
4313:
4311:
4308:
4307:
4292:
4264:
4249:
4234:
4219:
4217:
4214:
4213:
4181:
4177:
4173:
4168:
4154:
4149:
4135:
4130:
4112:
4108:
4104:
4099:
4097:
4094:
4093:
4087:
4057:
4042:
4027:
3981:
3970:
3967:
3966:
3881:
3870:
3867:
3866:
3828:
3825:
3824:
3805:
3775:
3760:
3745:
3743:
3740:
3739:
3659:
3648:
3645:
3644:
3628:If two events,
3626:
3602:
3597:
3594:
3593:
3574:
3569:
3566:
3565:
3519:
3517:
3514:
3513:
3482:
3480:
3477:
3476:
3440:
3439:
3418:
3417:
3415:
3392:
3390:
3387:
3386:
3344:
3342:
3339:
3338:
3323:
3317:
3281:
3278:
3277:
3227:
3212:
3201:
3198:
3197:
3194:
3182:
3161:
3137:
2975:
2949:
2923:
2751:
2746:
2740:
2737:
2736:
2694:But, later on:
2686:
2637:
2619:
2567:
2535:
2507:
2401:
2378:
2367:
2344:
2340:
2294:
2248:
2246:Interpretations
2214:
2210:
2206:
2201:
2169:
2145:
2116:
1990:—The preceding
1985:
1963:
1906:to the article
1840:
1795:
1785:122.162.135.235
1769:
1755:
1688:
1635:
1609:
1495:
1468:
1453:
1438:
1421:
1396:
1326:
1322:
1320:
1308:
1303:
1299:
1294:
1291:
1290:
1270:
1249:
1175:
1171:
1169:
1153:
1149:
1148:
1144:
1136:
1132:
1131:
1078:
1074:
1072:
1060:
1039:
1036:
1035:
1010:
987:
968:
952:
914:
911:
841:
837:
835:
823:
818:
814:
809:
806:
805:
785:
764:
690:
686:
684:
668:
664:
663:
659:
651:
647:
646:
593:
589:
587:
575:
554:
551:
550:
525:
502:
327:interpretations
263:
170:applied sense.
72:
59:
30:
22:
21:
20:
12:
11:
5:
6095:
6085:
6084:
6047:
6044:
6028:
6025:
6021:
6020:
5985:
5984:
5980:
5948:
5945:
5944:
5943:
5925:
5884:
5881:
5864:
5861:
5860:
5859:
5818:
5815:
5790:
5787:
5766:
5765:
5758:
5711:
5710:
5702:Added archive
5680:
5677:
5676:
5675:
5674:
5673:
5672:
5671:
5598:
5575:
5570:
5569:
5567:
5535:
5532:
5508:
5505:
5496:
5493:
5484:
5481:
5472:
5469:
5468:
5467:
5465:
5453:
5360:
5353:
5352:
5351:
5350:
5349:
5337:here, people!
5317:
5316:
5286:
5283:
5280:
5277:
5276:
5267:
5258:
5248:
5247:
5244:
5230:
5229:
5224:
5212:
5209:
5206:
5205:
5189:
5186:
5183:
5180:
5175:
5172:
5169:
5166:
5163:
5160:
5156:
5152:
5149:
5146:
5140:
5134:
5131:
5128:
5125:
5120:
5117:
5114:
5111:
5108:
5105:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5071:
5067:
5066:
5042:
5039:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
5013:
5011:
5008:
5005:
5002:
4999:
4996:
4993:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4970:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4945:
4942:
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4935:
4931:
4928:
4925:
4922:
4919:
4917:
4915:
4912:
4909:
4906:
4903:
4900:
4897:
4896:
4886:
4882:
4881:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4825:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4759:
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4753:
4750:
4748:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4727:
4717:
4713:
4712:
4699:
4696:
4693:
4690:
4687:
4684:
4681:
4678:
4673:
4669:
4665:
4662:
4652:
4648:
4647:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4597:
4593:
4592:
4589:
4579:
4576:
4563:
4543:
4523:
4520:
4516:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4481:
4477:
4474:
4471:
4447:
4425:
4421:
4398:
4394:
4384:. The odds on
4373:
4347:
4343:
4320:
4316:
4291:
4288:
4272:
4269:
4263:
4257:
4254:
4248:
4242:
4239:
4233:
4227:
4224:
4210:
4209:
4198:
4194:
4190:
4180:
4176:
4171:
4167:
4163:
4160:
4157:
4152:
4148:
4144:
4141:
4138:
4133:
4129:
4125:
4121:
4111:
4107:
4102:
4086:
4083:
4071:
4065:
4062:
4056:
4050:
4047:
4041:
4035:
4032:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3980:
3977:
3974:
3959:
3958:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3880:
3877:
3874:
3847:
3844:
3841:
3838:
3835:
3832:
3804:
3801:
3789:
3783:
3780:
3774:
3768:
3765:
3759:
3753:
3750:
3736:
3735:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3658:
3655:
3652:
3625:
3622:
3609:
3605:
3601:
3581:
3577:
3573:
3563:
3562:
3538:
3535:
3532:
3529:
3526:
3522:
3501:
3498:
3495:
3492:
3489:
3485:
3473:
3472:
3459:
3453:
3450:
3447:
3443:
3437:
3434:
3431:
3428:
3425:
3421:
3414:
3411:
3408:
3405:
3402:
3399:
3395:
3363:
3360:
3357:
3354:
3351:
3347:
3319:Main article:
3316:
3313:
3300:
3297:
3294:
3291:
3288:
3285:
3256:If two events
3235:
3232:
3226:
3220:
3217:
3211:
3208:
3205:
3163:
3162:
3159:
3158:
3136:
3133:
3132:
3131:
3130:
3129:
3128:
3127:
3126:
3125:
3124:
3123:
3100:
3099:
3098:
3097:
3096:
3095:
3094:
3093:
3072:
3071:
3070:
3069:
3068:
3067:
3048:
3047:
3046:
3045:
3030:
3013:
3012:
3011:
3010:
2948:
2945:
2922:
2919:
2918:
2917:
2901:
2900:
2895:193.109.51.150
2889:
2888:
2866:
2825:
2823:
2822:
2799:
2796:
2793:
2790:
2787:
2784:
2779:
2774:
2771:
2768:
2765:
2762:
2759:
2754:
2749:
2745:
2732:
2731:
2703:
2685:
2682:
2681:
2680:
2679:
2678:
2636:
2633:
2631:
2618:
2615:
2591:
2577:173.230.96.116
2566:
2563:
2534:
2531:
2455:
2454:
2400:
2397:
2366:
2363:
2339:
2336:
2335:
2334:
2318:
2293:
2290:
2289:
2288:
2274:
2273:
2247:
2244:
2243:
2242:
2212:
2208:
2204:
2200:
2197:
2196:
2195:
2168:
2165:
2164:
2163:
2137:
2136:
2115:
2112:
2111:
2110:
2101:citing sources
2061:
2034:
2033:
2032:
2031:
2019:
2018:
1984:
1981:
1980:
1979:
1962:
1959:
1958:
1957:
1956:
1955:
1954:
1953:
1952:
1951:
1950:
1949:
1839:
1836:
1835:
1834:
1794:
1791:
1768:
1765:
1759:BetacommandBot
1754:
1751:
1749:
1747:
1746:
1745:
1744:
1732:
1731:
1687:
1684:
1683:
1682:
1681:
1680:
1667:
1666:
1651:richardajohns
1634:
1631:
1608:
1605:
1604:
1603:
1584:
1509:
1494:
1491:
1490:
1489:
1467:
1464:
1452:
1449:
1437:
1434:
1420:
1417:
1416:
1415:
1395:
1392:
1391:
1390:
1382:
1381:
1376:
1375:
1374:
1373:
1362:
1359:
1355:
1351:
1348:
1345:
1342:
1339:
1334:
1329:
1325:
1317:
1314:
1311:
1307:
1302:
1298:
1285:
1284:
1266:
1245:
1237:
1221:
1220:
1207:
1204:
1200:
1196:
1193:
1189:
1183:
1178:
1174:
1168:
1165:
1162:
1159:
1156:
1152:
1147:
1143:
1140:
1100:
1099:
1086:
1081:
1077:
1069:
1066:
1063:
1059:
1055:
1052:
1049:
1046:
1043:
1008:
986:
983:
967:
964:
951:
948:
942:
941:
910:
907:
906:
905:
897:
896:
891:
890:
889:
888:
877:
874:
870:
866:
863:
860:
857:
854:
849:
844:
840:
832:
829:
826:
822:
817:
813:
800:
799:
781:
760:
752:
736:
735:
722:
719:
715:
711:
708:
704:
698:
693:
689:
683:
680:
677:
674:
671:
667:
662:
658:
655:
615:
614:
601:
596:
592:
584:
581:
578:
574:
570:
567:
564:
561:
558:
523:
501:
498:
497:
496:
495:
491:
490:
486:
485:
481:
480:
476:
475:
467:
466:
462:
461:
457:
456:
452:
451:
447:
446:
438:
437:
430:
429:
425:
424:
421:Bayes' theorem
412:
411:
407:
406:
402:
401:
393:
392:
388:
387:
383:
382:
377:
376:
372:
371:
367:
366:
349:
341:
340:
339:
291:
290:
261:
254:
246:
242:
217:
215:
206:
202:
196:
180:
160:
146:
133:
110:
106:
102:
92:
77:
71:
68:
65:
64:
52:
51:
34:
15:
9:
6:
4:
3:
2:
6094:
6083:
6079:
6075:
6074:MartinPoulter
6070:
6069:
6068:
6067:
6063:
6059:
6054:
6051:
6043:
6042:
6038:
6034:
6031:Set are what
6016:
6013:
6009:
6005:
6002:(1): 93–106.
6001:
5997:
5990:
5986:
5983:
5979:
5978:
5974:
5970:
5966:
5965:playing cards
5962:
5958:
5954:
5942:
5938:
5934:
5930:
5926:
5922:
5918:
5913:
5912:
5911:
5909:
5905:
5901:
5897:
5896:71.207.141.13
5893:
5880:
5879:
5875:
5871:
5858:
5855:
5853:
5849:
5848:
5842:
5839:
5835:
5834:
5833:
5832:
5828:
5824:
5814:
5813:
5809:
5805:
5801:
5797:
5786:
5785:
5780:
5775:
5774:
5763:
5759:
5756:
5752:
5751:
5750:
5743:
5737:
5733:
5729:
5725:
5719:
5714:
5709:
5705:
5701:
5700:
5699:
5697:
5693:
5689:
5684:
5670:
5666:
5662:
5659:
5655:
5654:
5653:
5649:
5645:
5641:
5637:
5633:
5632:
5631:
5627:
5623:
5619:
5615:
5611:
5607:
5603:
5599:
5596:
5592:
5591:
5590:
5589:
5585:
5581:
5574:
5568:
5565:
5558:
5554:
5550:
5544:
5540:
5539:Probabilicity
5534:Probabilicity
5531:
5528:
5524:
5520:
5514:
5507:Cultural uses
5504:
5500:
5492:
5490:
5480:
5476:
5466:
5464:
5461:
5457:
5449:
5448:
5447:
5445:
5442:
5438:
5434:
5427:, 7 July 2015
5422:
5418:
5414:
5407:
5405:
5402:
5398:
5394:
5387:, 7 July 2015
5382:
5378:
5374:
5368:
5367:
5358:
5348:
5344:
5340:
5336:
5335:summary style
5332:
5328:
5324:
5321:
5320:
5319:
5318:
5315:
5311:
5307:
5303:
5302:
5301:
5300:
5296:
5292:
5271:
5262:
5253:
5249:
5246:
5245:
5243:
5240:
5236:
5228:
5225:
5223:
5220:
5219:
5218:
5217:
5184:
5178:
5170:
5164:
5158:
5150:
5144:
5138:
5129:
5123:
5115:
5112:
5109:
5103:
5097:
5091:
5088:
5085:
5079:
5072:
5068:
5037:
5031:
5025:
5019:
5016:
5014:
5006:
5003:
5000:
4994:
4984:
4978:
4972:
4964:
4958:
4955:
4949:
4943:
4937:
4929:
4923:
4920:
4918:
4910:
4907:
4904:
4898:
4887:
4883:
4852:
4846:
4843:
4837:
4831:
4828:
4826:
4818:
4815:
4812:
4806:
4796:
4793:
4790:
4784:
4781:
4775:
4769:
4766:
4760:
4754:
4751:
4749:
4741:
4738:
4735:
4729:
4718:
4714:
4694:
4688:
4685:
4682:
4679:
4671:
4667:
4660:
4653:
4649:
4629:
4626:
4623:
4617:
4611:
4605:
4598:
4594:
4590:
4587:
4586:
4575:
4561:
4541:
4518:
4510:
4504:
4498:
4492:
4489:
4483:
4475:
4469:
4461:
4445:
4423:
4419:
4396:
4392:
4371:
4363:
4345:
4341:
4318:
4314:
4305:
4301:
4297:
4287:
4270:
4267:
4261:
4255:
4252:
4246:
4240:
4237:
4231:
4225:
4222:
4196:
4192:
4188:
4178:
4174:
4165:
4161:
4158:
4155:
4146:
4142:
4139:
4136:
4127:
4123:
4119:
4109:
4105:
4092:
4091:
4090:
4082:
4069:
4063:
4060:
4054:
4048:
4045:
4039:
4033:
4030:
4024:
4018:
4012:
4009:
4003:
3997:
3994:
3988:
3978:
3972:
3964:
3945:
3939:
3933:
3930:
3924:
3918:
3915:
3909:
3906:
3903:
3897:
3894:
3888:
3878:
3872:
3865:
3864:
3863:
3861:
3842:
3839:
3836:
3830:
3822:
3818:
3814:
3810:
3800:
3787:
3781:
3778:
3772:
3766:
3763:
3757:
3751:
3748:
3720:
3714:
3708:
3702:
3696:
3693:
3687:
3684:
3681:
3675:
3672:
3666:
3656:
3650:
3643:
3642:
3641:
3639:
3635:
3631:
3621:
3607:
3603:
3599:
3579:
3575:
3571:
3560:
3556:
3552:
3533:
3530:
3527:
3499:
3496:
3490:
3457:
3448:
3432:
3429:
3426:
3412:
3406:
3403:
3400:
3385:
3384:
3383:
3381:
3377:
3358:
3355:
3352:
3336:
3332:
3328:
3322:
3312:
3295:
3292:
3289:
3283:
3276:, denoted as
3275:
3271:
3267:
3263:
3259:
3254:
3252:
3233:
3230:
3224:
3218:
3215:
3209:
3206:
3203:
3190:
3186:
3180:
3176:
3172:
3168:
3157:
3155:
3151:
3147:
3143:
3122:
3118:
3114:
3110:
3109:
3108:
3107:
3106:
3105:
3104:
3103:
3102:
3101:
3092:
3088:
3084:
3080:
3079:
3078:
3077:
3076:
3075:
3074:
3073:
3066:
3062:
3058:
3054:
3053:
3052:
3051:
3050:
3049:
3044:
3040:
3036:
3031:
3029:
3025:
3021:
3017:
3016:
3015:
3014:
3009:
3005:
3001:
2997:
2996:
2995:
2994:
2993:
2991:
2987:
2983:
2982:68.44.139.107
2979:
2974:
2969:
2964:
2960:
2956:
2953:
2944:
2943:
2939:
2935:
2929:
2926:
2916:
2912:
2908:
2903:
2902:
2899:
2896:
2891:
2890:
2887:
2883:
2879:
2875:
2871:
2870:
2869:
2864:
2863:
2859:
2855:
2849:
2848:
2844:
2840:
2835:
2829:
2826:
2821:
2818:
2813:
2797:
2794:
2788:
2782:
2772:
2769:
2763:
2757:
2752:
2747:
2743:
2734:
2733:
2730:
2727:
2723:
2719:
2718:
2717:
2714:
2711:
2708:
2705:
2701:
2698:
2695:
2692:
2689:
2677:
2673:
2669:
2665:
2662:
2661:
2660:
2657:
2653:
2652:
2651:
2650:
2646:
2642:
2641:94.193.241.76
2632:
2629:
2628:
2625:
2614:
2611:
2607:
2603:
2597:
2594:
2587:
2586:
2582:
2578:
2572:
2571:
2562:
2560:
2556:
2555:
2552:
2546:
2543:presented by
2542:
2541:
2530:
2527:
2523:
2519:
2515:
2511:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2453:
2450:
2446:
2442:
2441:
2440:
2439:
2435:
2431:
2425:
2424:
2421:
2415:
2414:
2411:
2407:
2396:
2394:
2390:
2386:
2382:
2376:
2372:
2362:
2360:
2356:
2352:
2351:86.166.148.24
2348:
2333:
2329:
2325:
2324:71.116.162.54
2321:
2320:
2319:
2316:
2315:
2312:
2309:
2306:
2301:
2297:
2287:
2284:
2280:
2276:
2275:
2272:
2269:
2265:
2264:
2263:
2262:
2258:
2254:
2241:
2237:
2233:
2229:
2224:
2223:
2222:
2221:
2218:
2194:
2189:
2188:
2187:
2186:
2182:
2178:
2173:
2162:
2159:
2155:
2151:
2142:
2141:
2140:
2135:
2131:
2127:
2122:
2121:
2120:
2109:
2106:
2102:
2098:
2094:
2090:
2086:
2081:
2080:
2079:
2078:
2074:
2070:
2066:
2059:
2058:
2055:
2051:
2047:
2043:
2040:
2037:
2030:
2027:
2023:
2022:
2021:
2020:
2017:
2014:
2010:
2009:
2008:
2005:
2001:
1997:
1996:203.129.46.17
1993:
1978:
1977:
1974:
1970:
1965:
1964:
1948:
1945:
1941:
1940:
1939:
1935:
1931:
1926:
1925:
1924:
1921:
1917:
1913:
1912:verifiability
1909:
1905:
1901:
1897:
1893:
1892:
1891:
1887:
1883:
1878:
1877:
1876:
1873:
1869:
1865:
1864:
1863:
1862:
1858:
1854:
1849:
1846:
1843:
1833:
1829:
1825:
1820:
1819:
1818:
1817:
1814:
1810:
1808:
1805:
1799:
1790:
1789:
1786:
1782:
1781:Almost Surely
1778:
1774:
1764:
1763:
1760:
1750:
1743:
1740:
1736:
1735:
1734:
1733:
1730:
1727:
1723:
1719:
1715:
1711:
1707:
1706:
1705:
1704:
1701:
1697:
1693:
1679:
1676:
1671:
1670:
1669:
1668:
1665:
1662:
1658:
1654:
1653:
1652:
1650:
1647:
1646:Richardajohns
1643:
1639:
1630:
1629:
1626:
1625:Siobhan Hansa
1623:. Thanks --
1622:
1616:
1614:
1602:
1599:
1595:
1590:
1586:
1585:
1583:
1581:
1576:
1574:
1570:
1566:
1561:
1556:
1554:
1551:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1508:
1507:
1504:
1500:
1488:
1485:
1480:
1479:
1478:
1477:
1474:
1473:24.80.119.229
1463:
1462:
1459:
1448:
1447:
1444:
1433:
1432:
1429:
1425:
1414:
1411:
1407:
1406:
1405:
1404:
1401:
1388:
1384:
1383:
1378:
1377:
1360:
1357:
1353:
1346:
1337:
1332:
1327:
1323:
1309:
1300:
1289:
1288:
1287:
1286:
1282:
1281:almost surely
1278:
1274:
1269:
1265:
1261:
1257:
1253:
1248:
1244:
1240:
1239:
1238:
1235:
1233:
1228:
1225:
1205:
1202:
1198:
1194:
1191:
1187:
1181:
1176:
1172:
1166:
1160:
1150:
1145:
1141:
1138:
1130:
1129:
1128:
1126:
1122:
1118:
1114:
1109:
1105:
1084:
1079:
1075:
1061:
1053:
1047:
1034:
1033:
1032:
1030:
1026:
1022:
1018:
1014:
1007:
1003:
999:
994:
990:
982:
981:
978:
973:
963:
961:
957:
947:
940:
936:
932:
927:
924:
922:
917:
903:
899:
898:
893:
892:
875:
872:
868:
861:
852:
847:
842:
838:
824:
815:
804:
803:
802:
801:
797:
796:almost surely
793:
789:
784:
780:
776:
772:
768:
763:
759:
755:
754:
753:
750:
748:
743:
740:
720:
717:
713:
709:
706:
702:
696:
691:
687:
681:
675:
665:
660:
656:
653:
645:
644:
643:
641:
637:
633:
629:
624:
620:
599:
594:
590:
576:
568:
562:
549:
548:
547:
545:
541:
537:
533:
529:
522:
518:
514:
509:
505:
493:
492:
488:
487:
483:
482:
478:
477:
473:
469:
468:
464:
463:
459:
458:
454:
453:
449:
448:
444:
440:
439:
435:
434:
433:
427:
426:
422:
418:
414:
413:
409:
408:
404:
403:
399:
395:
394:
390:
389:
385:
384:
379:
378:
374:
373:
369:
368:
364:
363:
362:
360:
355:
348:
346:
337:
332:
328:
323:
322:
321:
317:
315:
309:
307:
302:
298:
296:
289:
288:
282:
280:
275:
269:
267:
258:
253:
251:
245:
241:
240:
236:
235:by definition
231:
226:
222:
216:
213:
209:
204:
200:
195:
194:
189:
185:
179:
175:
171:
167:
163:
159:
158:
152:
151:
145:
144:
139:
132:
131:
126:
122:
118:
115:
109:
105:
100:
96:
91:
90:
86:
81:
76:
62:
58:
57:
49:
45:
41:
40:
35:
28:
27:
19:
6055:
6052:
6049:
6033:72.27.101.84
6030:
5999:
5995:
5989:
5981:
5952:
5950:
5890:— Preceding
5886:
5866:
5845:
5844:
5823:173.66.2.216
5820:
5792:
5770:
5767:
5742:source check
5721:
5715:
5712:
5685:
5682:
5602:User:Edwardx
5594:
5577:
5566:
5547:— Preceding
5542:
5538:
5537:
5517:— Preceding
5510:
5501:
5498:
5486:
5477:
5474:
5450:
5431:— Preceding
5429:
5417:2.84.216.225
5411:— Preceding
5408:
5391:— Preceding
5389:
5377:2.84.216.225
5371:— Preceding
5365:
5364:
5362:
5357:one and only
5356:
5288:
5270:
5261:
5252:
5232:
5231:
5226:
5221:
5215:
5214:
4591:Probability
4459:
4362:conditioning
4302:relates the
4293:
4211:
4088:
3960:
3820:
3816:
3812:
3808:
3806:
3737:
3633:
3629:
3627:
3549:is formally
3474:
3379:
3375:
3334:
3330:
3329:of an event
3324:
3273:
3269:
3261:
3257:
3255:
3188:
3184:
3178:
3174:
3173:of an event
3170:
3166:
3164:
3138:
2976:— Preceding
2970:
2965:
2961:
2957:
2954:
2950:
2930:
2927:
2924:
2878:Double sharp
2865:
2850:
2830:
2827:
2824:
2811:
2715:
2712:
2709:
2706:
2702:
2699:
2696:
2693:
2690:
2687:
2668:50.0.136.106
2638:
2630:
2620:
2606:31.51.15.214
2600:— Preceding
2588:
2573:
2569:
2568:
2549:
2545:Melvyn Bragg
2539:
2536:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2426:
2416:
2402:
2385:99.36.94.205
2368:
2341:
2317:
2303:
2299:
2295:
2249:
2202:
2174:
2170:
2167:Exclusive Or
2154:Exclusive or
2150:Inclusive or
2138:
2117:
2084:
2060:
2052:
2048:
2044:
2041:
2038:
2035:
1986:
1966:
1903:
1867:
1850:
1847:
1844:
1841:
1800:
1796:
1776:
1772:
1770:
1756:
1748:
1689:
1644:
1640:
1636:
1617:
1610:
1577:
1569:sample space
1564:
1559:
1557:
1540:
1537:propositions
1536:
1528:
1520:
1516:
1513:Frequentists
1510:
1496:
1469:
1454:
1439:
1422:
1397:
1276:
1272:
1267:
1263:
1251:
1246:
1242:
1236:
1229:
1226:
1222:
1124:
1120:
1116:
1112:
1107:
1103:
1101:
1028:
1020:
1016:
1012:
1005:
1001:
997:
995:
991:
988:
969:
953:
945:
912:
791:
787:
782:
778:
766:
761:
757:
751:
744:
741:
737:
639:
635:
631:
627:
622:
618:
616:
543:
535:
531:
527:
520:
516:
512:
510:
506:
503:
431:
350:
342:
330:
326:
318:
313:
310:
305:
303:
299:
294:
292:
283:
278:
273:
270:
265:
259:
255:
249:
247:
243:
234:
229:
224:
221:epistemology
218:
214:
210:
205:
201:
197:
187:
183:
181:
176:
172:
168:
164:
161:
153:
148:
147:
137:
134:
124:
113:
111:
107:
101:
97:
93:
84:
82:
78:
73:
60:
43:
37:
6027:What is set
5951:I added...
5688:Probability
4300:Bayes' rule
3823:denoted as
3638:independent
2540:In Our Time
2533:In Our Time
2508:—Preceding
2379:—Preceding
2345:—Preceding
1908:Probability
1739:MisterSheik
1718:Probability
1611:This link:
1594:Probability
1529:experiments
1503:MisterSheik
1410:The Infidel
1400:The Infidel
977:Christianjb
36:This is an
5982:References
5779:Report bug
5606:Improbable
5595:Improbable
5553:2.84.211.1
5437:2.84.211.1
5397:2.84.211.1
5366:probabilon
5211:References
5070:A given B
3187:) = 1 - P(
3171:complement
3154:WP:SUMMARY
3083:LadyLeodia
2947:Definition
2635:Likelihood
2554:Farmbrough
2177:Joeldipops
2075:#REDIRECT
2054:JamesDowty
1824:Salix alba
1642:evidence.
1621:guidelines
1613:http://www
1560:statements
1545:subjective
1387:Ben Cairns
1108:assumption
931:Salix alba
902:Ben Cairns
623:assumption
274:impossible
150:TedDunning
6015:144540412
5929:this edit
5923:. Thanks.
5870:BrianH123
5804:Lbeaumont
5762:this tool
5755:this tool
5610:User:czar
5573:SpatialOS
5489:pippo2001
5285:Responses
4411:to event
4333:to event
4306:of event
3811:or event
3555:σ-algebra
3551:undefined
3152:) as per
2408:. Thanks
1967:Moved to
1541:uncertain
1539:that are
1533:Bayesians
1519:that are
1424:certainty
1419:certainty
445:later on.
336:AxelBoldt
287:AxelBoldt
61:Archive 1
6058:Ajosephg
5953:Gambling
5904:contribs
5892:unsigned
5768:Cheers.—
5661:sheridan
5614:sheridan
5580:sheridan
5549:unsigned
5519:unsigned
5483:Good Job
5433:unsigned
5413:unsigned
5393:unsigned
5373:unsigned
4885:A and B
3378:, given
3167:opposite
2978:unsigned
2834:infinity
2656:~Asarlaí
2602:unsigned
2522:contribs
2514:Aklinger
2510:unsigned
2420:Ben Finn
2381:unsigned
2347:unsigned
2253:Melcombe
2004:contribs
1992:unsigned
1675:Ben Finn
1657:Be Bold!
1523:, i.e.,
1466:Entropy?
1428:Spencerk
1104:a priori
950:Untitled
619:a priori
95:theory.
70:Untitled
5915:Please
5692:my edit
5656:Thanks
5423:) 01:38
5383:) 01:38
4716:A or B
2907:Mmart71
2726:Zundark
2596:Unknown
2593:Unknown
2449:Lambiam
2430:Krantz2
2410:Andeggs
2292:Aliens?
2283:Lambiam
2268:Lambiam
2230:.) --
2217:Lambiam
2193:Lambiam
2158:Lambiam
2105:Lambiam
2065:Krantz2
1973:Lambiam
1944:Lambiam
1920:Lambiam
1898:or the
1872:Lambiam
1813:Lambiam
1696:Lambiam
1661:Andeggs
996:... As
511:... As
125:explain
39:archive
5847:Tayste
5513:Person
5452:edits.
3249:. See
3183:P(not
3057:Nijdam
2624:Wingda
2371:Dbjohn
2232:Avenue
2226:book:
2013:Avenue
1777:almost
1773:almost
1690:Since
1521:random
1517:events
1258:, the
1241:Since
1017:define
1002:expect
773:, the
756:Since
532:define
517:expect
331:models
87::-) --
6012:S2CID
5852:edits
5616:. —
5456:Jerzy
5235:Jerzy
4651:not A
4588:Event
3512:then
2720:In a
2311:Drake
2308:Jason
2305:Brian
2281:. --
2156:. --
2103:. --
2036:Hi,
1493:Moved
1458:Zalle
1443:Zalle
1380:edit!
1192:: -->
1123:: -->
1027:, as
895:edit!
707:: -->
638:: -->
542:, as
306:guess
279:every
225:apply
85:that.
16:<
6078:talk
6062:talk
6037:talk
5973:talk
5961:dice
5937:talk
5933:DVdm
5917:sign
5900:talk
5883:Misc
5874:talk
5840:and
5838:dice
5827:talk
5808:talk
5665:talk
5584:talk
5557:talk
5527:talk
5495:Luck
5441:talk
5421:talk
5401:talk
5381:talk
5363:the
5343:talk
5310:talk
5306:Bhny
5295:talk
4304:odds
4184:and
3819:and
3662:and
3636:are
3632:and
3325:The
3272:and
3260:and
3165:The
3117:talk
3113:Bhny
3087:talk
3061:talk
3039:talk
3035:Bhny
3024:talk
3020:Bhny
3004:talk
3000:Bhny
2986:talk
2938:talk
2911:talk
2882:talk
2872:See
2858:talk
2843:talk
2817:Wake
2672:talk
2645:talk
2610:talk
2581:talk
2551:Rich
2518:talk
2434:talk
2406:here
2389:talk
2375:talk
2355:talk
2328:talk
2277:See
2257:talk
2236:talk
2181:talk
2152:and
2130:talk
2085:your
2069:talk
2000:talk
1934:talk
1886:talk
1857:talk
1828:talk
1783:? --
1726:iNic
1598:iNic
1484:iNic
1232:Tlee
1203:<
935:talk
747:Tlee
718:<
329:and
188:fact
184:what
6004:doi
5736:RfC
5706:to
5561::16
5425::55
5385::18
4294:In
4115:or
3984:or
3965:is
3963:die
3884:or
3561:).
3475:If
3268:of
3169:or
2026:JJL
1306:lim
1111:Pr(
1058:lim
821:lim
626:Pr(
573:lim
250:why
239:LDC
193:LMS
157:LDC
143:LDC
130:LMS
89:LMS
6080:)
6064:)
6039:)
6010:.
6000:11
5998:.
5975:)
5939:)
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5902:•
5876:)
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5829:)
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5749:.
5744:}}
5740:{{
5667:)
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5646:·
5628:)
5624:·
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5454:--
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5312:)
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5233:--
5113:∩
5089:∣
5004:∩
4908:∩
4816:∪
4794:∩
4782:−
4739:∪
4686:−
4618:∈
4490:∝
4462:,
4271:26
4268:11
4256:52
4247:−
4241:52
4238:12
4226:52
4223:13
4166:−
3907:∪
3840:∪
3758:×
3685:∩
3531:∣
3430:∩
3404:∣
3356:∣
3311:.
3293:∩
3210:−
3148:,
3144:,
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2778:⟹
2744:∫
2674:)
2666:.
2647:)
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2557:,
2524:)
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2207:+x
2183:)
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2006:).
2002:•
1936:)
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1868:my
1859:)
1830:)
1822:--
1659:.
1575:.
1555:.
1552::
1385:--
1361:1.
1341:Pr
1316:∞
1313:→
1297:Pr
1230:--
1206:ϵ
1195:δ
1167:−
1155:Pr
1127:,
1068:∞
1065:→
1042:Pr
937:)
900:--
876:1.
856:Pr
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828:→
812:Pr
745:--
721:ϵ
710:δ
682:−
670:Pr
642:,
583:∞
580:→
557:Pr
381:2.
266:is
230:is
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141:--
6076:(
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6017:.
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5185:B
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5092:B
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5035:(
5032:P
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5007:B
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4965:B
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4930:A
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4911:B
4905:A
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4853:B
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4844:+
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4838:A
4835:(
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4819:B
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4807:P
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4776:B
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4758:(
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4752:=
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4742:B
4736:A
4733:(
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4689:P
4683:1
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4672:c
4668:A
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4633:]
4630:1
4627:,
4624:0
4621:[
4615:)
4612:A
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4562:B
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4519:A
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4508:(
4505:P
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4476:A
4473:(
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4424:2
4420:A
4397:1
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4372:B
4346:2
4342:A
4319:1
4315:A
4262:=
4253:3
4232:+
4197:.
4193:)
4189:B
4179:A
4175:(
4170:P
4162:)
4159:B
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4151:P
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4140:A
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4132:P
4128:=
4124:)
4120:B
4110:A
4106:(
4101:P
4070:.
4064:3
4061:1
4055:=
4049:6
4046:1
4040:+
4034:6
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4025:=
4022:)
4019:2
4016:(
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4010:+
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4004:1
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3998:P
3995:=
3992:)
3989:2
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3943:)
3940:B
3937:(
3934:P
3931:+
3928:)
3925:A
3922:(
3919:P
3916:=
3913:)
3910:B
3904:A
3901:(
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3889:B
3879:A
3876:(
3873:P
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3837:A
3834:(
3831:P
3821:B
3817:A
3813:B
3809:A
3788:.
3782:4
3779:1
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3767:2
3764:1
3752:2
3749:1
3721:,
3718:)
3715:B
3712:(
3709:P
3706:)
3703:A
3700:(
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3694:=
3691:)
3688:B
3682:A
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3676:P
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3670:)
3667:B
3657:A
3654:(
3651:P
3634:B
3630:A
3608:3
3604:/
3600:1
3580:2
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3572:1
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3534:B
3528:A
3525:(
3521:P
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3497:=
3494:)
3491:B
3488:(
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3458:.
3452:)
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3427:A
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3410:)
3407:B
3401:A
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3353:A
3350:(
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3296:B
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3258:A
3234:6
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3219:6
3216:1
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3189:A
3185:A
3179:A
3175:A
3115:(
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1333:N
1328:H
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1080:H
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1054:=
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1048:H
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1011:/
1009:H
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848:N
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767:N
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762:H
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666:|
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657:r
654:P
640:n
636:N
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628:H
600:N
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591:N
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569:=
566:)
563:H
560:(
544:N
536:H
528:N
526:/
524:H
521:N
513:N
262:1
50:.
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