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Topic: Probability axioms

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	Probability - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		In Cox's formulation, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions.
		Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events.
		Governments typically apply probability methods in environmental regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole.
	en.wikipedia.org /wiki/Probability (2765 words)

	Probability axioms - Wikipedia, the free encyclopedia
		The probability P of some event E, denoted P(E), is defined with respect to a "universe", or sample space Ω, of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms.
		Alternatively, a probability can be interpreted as a measure on a σ-algebra of subsets of the sample space, those subsets being the events, such that the measure of the whole set equals 1.
		If the conditional probability of B given A is the same as the probability of B, then A and B are said to be independent.
	en.wikipedia.org /wiki/Probability_axioms (488 words)

	20th WCP: Coherence and Epistemic Rationality
		The claim that degrees of confidence should satisfy the probability axioms is most often defended by appealing to the so-called Dutch Book argument, which was first presented by Ramsey in his famous paper "Truth and Probability".
		Regardless of how the objective probabilities to which our degrees of confidence should ideally correspond are interpreted, the probability for necessary truths will be one and that of their denials will be zero.
		Probability functions are defined over Boolean algebras of propositions, where an assignment of probability one is required of those propositions with a particular structural role in the algebra.
	www.bu.edu /wcp/Papers/TKno/TKnoVine.htm (3166 words)

	Standard Probability Axioms
		Normally or at times, we think that it is possible or likely or probable that we may possess knowledge that is uncertain, that is, knowledge to which we are unable to assign the value of true or false....but nonetheless think that we can assign varying degrees of certainty to that knowledge.
		The theory of probabilities is the result, and one proposal is that probability theory provides the appropriate treatment of all of the various types of uncertain knowledge that we may entertain.
		On this view the semantics human of uncertainty is equivalent to probabilities; that is, for each uncertain belief that a person holds, there is some "subjective probability" that represents the degree to which the person believes the statement to be true.
	www.rci.rutgers.edu /~cfs/305_html/Induction/ProbabilityAxioms.html (1023 words)

	UCLA Soc. 210A, Topic 4, Probability
		One way of giving the probability axioms empirical referents is to imagine a long sequence of "trials" under identical conditions, each of which might or might not produce the event whose probability is under discussion.
		Probabilities are variously stated as decimal fractions, percents, or common fractions, and the latter sometimes have numerator and denominator separated by the word "in" instead of a horizontal bar or slash.
		The idea is that collecting additional information could move the actor's subjective probability away from the critical point at which the decision could go either way, thereby increasing his or her degree of certainty that the decision being made (to trust or not) is the correct one.
	www.sscnet.ucla.edu /soc/faculty/mcfarland/soc210a/prob.htm (5556 words)

	[No title]
		The Axioms can be used together to find a formula for the probability of a union of two events that are not necessarily disjoint in terms of the probability of each of the events and the probability of their intersection.
		The Axioms of Probability are mathematical rules that must be followed in assigning probabilities to events: The probability of an event cannot be negative, the probability that something happens must be 100%, and if two events cannot both occur, the probability that either occurs is the sum of the probabilities that each occurs.
		The updated probability is the conditional probability of A given B, which is equal to the probability that A and B both occur, divided by the probability that B occurs, provided that the probability that B occurs is not zero.
	www.stat.berkeley.edu /users/stark/SticiGui/Text/ch9.htm (4729 words)

	Wikinfo \| Probability axioms
		Alternatively, a probability can be interpreted as a measure on a sigma-algebra of subsets of the sample space, those subsets being the events, such that the measure of the whole set equals 1.
		B and A are said to be independent if the conditional probability of B given A is the same as the probability of B. In the case that the sample space is finite or countably infinite, a probability function can also be defined by its values on the elementary events e1, e2,...
		That is, the probability that some elementary event in the entire sample set will occur is 1, or certainty.
	www.wikinfo.org /wiki.php?title=Probability_axioms (525 words)

	Classical Probability Theory and Learning from Experience
		That is: The conditional probability of Q, given or assumed that P is true, equals the probability that (PandQ) is true, divided by the probability that (P) is true.
		The probability of a theory increases as its competing theories are falsified.
		The probability of a theory is proportional to the probability of its predictions.
	www.xs4all.nl /~maartens/logic/CPTandLFE.htm (4177 words)

	Probability Measure
		Axioms 1 and 2 are really just a matter of convention; we choose to measure the probability of an event with a number between 0 and 1 (as opposed, say, to a number between 0 and 100).
		Intuitively, the probability of an event is supposed to measure the long-term relative frequency of the event.
		In particular, the inclusion-exclusion rule is as important in combinatorics (the study of counting measure) as it is in probability.
	www.fmi.uni-sofia.bg /vesta/Virtual_Labs/prob/prob3.html (786 words)

	Interpretations of Probability
		This axiom is the cornerstone of the assimilation of probability theory to measure theory.
		Probability is thought of as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome.
		The laws of probability then are claimed to be constraints on these estimates: putative necessary conditions for minimizing her ‘losses’ in a broad sense, be they monetary, or measured by distances from the assignments of these experts.
	plato.stanford.edu /entries/probability-interpret (15179 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The mathematical aspect, described in Chapter 9, "Probability: Axioms and Fundaments," starts with a collection of axioms (assumptions) and derives consequences that are true for anything that satisfies the axioms.
		In the Frequency Theory of Probability, probability is the limit of the relative frequency with which an event occurs in repeated (independent--more about this in Chapter 9, Probability: Axioms and Fundaments) trials.
		In the Subjective Theory, evidence against the hypothesis that "the probability that A occurs is p%" is found by psychological testing to see whether the individual making the statement is telling the truth and is internally consistent in his assignments of probability.
	www.stat.berkeley.edu /users/stark/SticiGui/Text/ch8.1.htm (2714 words)

	A probability measure P(?) is a set function that assigns to any event E the real number P(E) called the probability of ... (Site not responding. Last check: 2007-11-07)
		A probability measure P(?) is a set function that assigns to any event E the real number P(E) called the probability of E which is a precise measure of the likelihood of the event E occurring
		Probability axioms are assumptions that tell us the "rules of the game" for the mathematics associated with probability.
		You may also think of these axioms as "truths" upon which we base all the mathematical derivations related to probability.
	www.pitt.edu /~jrclass/e20/notes/OH2.html (179 words)

	Probability 'Cepts
		probability theory is a central component of statistics.
		Given its centrality to statistics and its usefulness to planners in addition to statistics, the lack of emphasis on probability in planning education is odd.
		This discussion of probability concepts owes a great deal to (in other words, it's unabashedly scribed from) a course on probability theory and its applications taught at Cornell University by Ward Whitt during the summer of 1966.
	www.uri.edu /cels/cpla/marsh/cpl526/probabilityCepts.htm (1340 words)

	Certain Doubts » Arbitrary Actions and Arbitrary Beliefs (Site not responding. Last check: 2007-11-07)
		Jon’s right about the clash between the axioms of probability being in conflict with the basic idea of rational acceptance, which is not to say that we cannot toy with probability models to make them look something like a model of rational acceptance.
		The clash is rooted in properties deep in the calculus, namely general properties inherited by probability being a measure on a ring; so Kyburg was right to finger the rule of adjunction (aggregation) as the source of troubles rather than deductive closure in general.
		But if you take enough propositions that have a high probability (and are to some degree independent) and also satisfy the other necessary conditions, whatever they are, for being justified and conjoin them together, you’ll get a conjunction that doesn’t have a high enough probability to be justified.
	bengal-ng.missouri.edu /~kvanvigj/certain_doubts/?p=487 (6021 words)

	Probability axioms
		) a different value there are nevertheless certain axioms which should always hold for internal consistency.
		These are the axioms of probability theory (which can be
		Sometimes this is called the 4th axiom, but it follows from the others.
	www.dcs.qmw.ac.uk /~norman/BBNs/Probability_axioms.htm (111 words)

	Forums - Why EMH is flawed: it relies on Probability Theory which CAN'T DEFINE Randomness (Site not responding. Last check: 2007-11-07)
		So Probability concept, which is purely mathematical, is not enough to define the concept of randomness.
		So Kolmogorov, the father of probability axioms in 1933, recognised their limit himself: the use of these axioms only relie on logical correctness - which is purely mathematical - not on their relevance to physical phenomena.
		Shewart already said far before anybody that the concept of randomness was physical and that one has to chose the one that is suitable to the context of use.
	www.elitetrader.com /vb/showthread.php?threadid=26293 (1128 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		the probability of testing positive given that you have the disease is 0.99, as is the probability of testing negative given that you dont have the disease.
		Suppose that you are a political prisoner in Russia and are to be exiled in one of two places: Siberia or Mongolia.
		It is also known that if you randomly select a person in Siberia, the probability that he will be wearing a seal-skin coat is 0.8, whereas the same event has probability 0.4 in Mongolia.
	www.cs.umu.se /kurser/TDBC70/HT00/grupp2.html (270 words)

	STA104 Homework 2: Axioms of Probability (Site not responding. Last check: 2007-11-07)
		Text Problems are from Sheldon Ross, A First Course in Probability (6th edn).
		NOTE: Remember that in probability the phrase "A or B" means that at least one of A and B occurs.
		This section of Statistics 104/Math 135 has thirty-nine students enrolled; if everyone has a birthday that is equally likely to be any of the 365 days of (non-leap) years, what is the probability that at least one student in the class has the same birthday as the instructor (me)?
	www.stat.duke.edu /courses/Fall03/sta104/hw/hw02.html (82 words)

	Axioms of Probability (Site not responding. Last check: 2007-11-07)
		Probability assigns to each event A a number
		Probabilities can express different opinions, differ among people.
		Probability is useful for expressing opinions on non-repeatable experiments:
	www.stat.umn.edu /~luke/classes/3091-3/notes/node11.html (172 words)

	James Worrell: Axioms for probability and nondeterminism (Site not responding. Last check: 2007-11-07)
		The former is modelled using the probabilistic powerdomain of Jones and Plotkin, while the latter is modelled by a geometrically convex variant of the Plotkin powerdomain.
		The main result is to show that the expected laws for probability and nondeterminism are sound and complete with respect to the model.
		We also present an operational semantics for the process algebra, and we show that the domain model is fully abstract with respect to probabilistic bisimilarity.
	www.math.tulane.edu /~jbw/express.html (110 words)

	Rational betting: the Dutch Book theorem (Site not responding. Last check: 2007-11-07)
		In this way, that of satisfying standard axioms for probability is proven to be a necessary condition for
		satisfies (P1-P2) then it identifies fair betting quotients, the standard probability functions actually provide a very elegant mathematical characterization of rational belief: An agents' beliefs are rational if and only if his degrees of belief are represented by a function satisfying the standard probability axioms.
		This actually explains what it is meant by saying that obeying the standard axioms of probability theory is a necessary (and sufficient) condition for rationality.
	www3.humnet.unipi.it /epistemologia/hosni/thesis/tesi/node86.html (341 words)

	Lecture Schedule (Site not responding. Last check: 2007-11-07)
		Lecture 4: (9/4) Review probability axioms, combinatorial analysis, finite and equally likely elementary events, multiplication principle, sampling (order relevant) with and without replacement, permutations, examples.
		Lecture 6: (9/9) Conditional probability definition, conditional probability axioms, Theorem of total probability, Bayes formula, examples.
		conditional probability axioms, Theorem of total probability, Bayes formula, examples, multiplication principle, examples, proability trees.
	www-ee.eng.hawaii.edu /~kuh/ee342.f02/lectures.html (773 words)

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