probability for the mathematical field of probability specifically rather than a general discussion see probability theory for other uses see probability disambiguation probability outline catalog of articles probabilists glossary notation journals category vte part of a series oncertainty approximation belief certainty doubt determinism fallibilism fatalism hypothesis justification nihilism proof scientific theory skepticism solipsism theory truth uncertainty related concepts and fundamentals agnosticism epistemology presupposition probability vte probability is the measure of the likelihood that an event will occur the higher the probability of an event the more likely it is that the event will occur a simple example is the tossing of a fair unbiased coin since the coin is fair the two outcomes heads and tails are both equally probable the probability of heads equals the probability of tails and since no other outcomes are possible the probability of either heads or tails is 12 which could also be written as 05 or 50 these concepts have been given an axiomatic mathematical formalization in probability theory which is used widely in such areas of study as mathematics statistics finance gambling science in particular physics artificial intelligencemachine learning computer science game theory and philosophy to for example draw inferences about the expected frequency of events probability theory is also used to describe the underlying mechanics and regularities of complex systems interpretations probability interpretations when dealing with experiments that are random and welldefined in a purely theoretical setting like tossing a fair coin probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes for example tossing a fair coin twice will yield headhead headtail tailhead and tailtail outcomes the probability of getting an outcome of headhead is 1 out of 4 outcomes or in numerical terms 14 025 or 25 however when it comes to practical application there are two major competing categories of probability interpretations whose adherents possess different views about the fundamental nature of probability objectivists assign numbers to describe some objective or physical state of affairs the most popular version of objective probability is frequentist probability which claims that the probability of a random event denotes the relative frequency of occurrence of an experiments outcome when repeating the experiment this interpretation considers probability to be the relative frequency in the long run of outcomes a modification of this is propensity probability which interprets probability as the tendency of some experiment to yield a certain outcome even if it is performed only once subjectivists assign numbers per subjective probability ie as a degree of belief etymology see also history of probability etymology further information likelihood the word probability derives from the latin probabilitas which can also mean probity a measure of the authority of a witness in a legal case in europe and often correlated with the witnesss nobility in a sense this differs much from the modern meaning of probability which in contrast is a measure of the weight of empirical evidence and is arrived at from inductive reasoning and statistical inference history history of probability the scientific study of probability is a modern development of mathematics gambling shows that there has been an interest in quantifying the ideas of probability for millennia but exact mathematical descriptions arose much later there are reasons for the slow development of the mathematics of probability whereas games of chance provided the impetus for the mathematical study of probability fundamental issues christiaan huygens likely published the first book on probability according to richard jeffrey before the middle of the seventeenth century the term probable latin probabilis meant approvable and was applied in that sense unequivocally to opinion and to action a probable action or opinion was one such as sensible people would undertake or hold in the circumstances gerolamo cardano the sixteenth century italian polymath gerolamo cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes aside from the elementary work by cardano the doctrine of probabilities dates to the correspondence of pierre de fermat and blaise pascal 1654 christiaan huygens 1657 gave the earliest known scientific treatment of the subject for histories of the early development of the very concept of mathematical probability the theory of errors may be traced back to roger cotess opera miscellanea posthumous 1722 but a memoir prepared by thomas simpson in 1755 printed 1756 first applied the theory to the discussion of errors of observation the reprint 1757 of this memoir lays down the axioms that positive and negative errors are equally probable and that certain assignable limits define the range of all errors simpson also discusses continuous errors and describes a probability curve the first two laws of error that were proposed both originated with pierresimon laplace the first law was published in 1774 and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error disregarding sign the second law of error was proposed in 1778 by laplace and stated that the frequency of the error is an exponential function of the square of the error daniel bernoulli 1778 introduced the principle of the maximum product of the probabilities of a system of concurrent errors carl friedrich gauss adrienmarie legendre 1805 developed the method of least squares and introduced it in his nouvelles mthodes pour la dtermination des orbites des comtes new methods for determining the orbits of comets in ignorance of legendres contribution an irishamerican writer robert adrain editor of the analyst 1808 first deduced the law of facility of error x c e h 2 x 2 displaystyle phi xceh2x2 where h displaystyle h is a constant depending on precision of observation and c displaystyle c is a scale factor ensuring that the area under the curve equals 1 he gave two proofs the second being essentially the same as john herschels 1850 for r the probable error of a single observation is well known in the nineteenth century authors on the general theory included laplace sylvestre lacroix 1816 littrow 1833 adolphe quetelet 1853 richard dedekind 1860 helmert 1872 hermann laurent 1873 liagre didion and karl pearson augustus de morgan and george boole improved the exposition of the theory andrey markov introduced on the geometric side see integral geometry contributors to the educational times were influential miller crofton mccoll wolstenholme watson and artemas martin further information history of statistics theory probability theory like other theories the theory of probability is a representation of its concepts in formal termsthat is in terms that can be considered separately from their meaning these formal terms are manipulated by the rules of mathematics and logic and any results are interpreted or translated back into the problem domain there have been at least two successful attempts to formalize probability namely the kolmogorov formulation and the cox formulation in kolmogorovs formulation see probability space sets are interpreted as events and probability itself as a measure on a class of sets in coxs theorem 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 in both cases the laws of probability are the same except for technical details there are other methods for quantifying uncertainty such as the dempstershafer theory or possibility theory but those are essentially different and not compatible with the laws of probability as usually understood applications probability theory is applied in everyday life in risk assessment and modeling the insurance industry and markets use actuarial science to determine pricing and make trading decisions governments apply probabilistic methods in environmental regulation entitlement analysis reliability theory of aging and longevity and financial regulation a good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread middle east conflict on oil prices which have ripple effects in the economy as a whole an assessment by a commodity trader that a war is more likely can send that commoditys prices up or down and signals other traders of that opinion accordingly the probabilities are neither assessed independently nor necessarily very rationally the theory of behavioral finance emerged to describe the effect of such groupthink on pricing on policy and on peace and conflict in addition to financial assessment probability can be used to analyze trends in biology eg disease spread as well as ecology eg biological punnett squares as with finance risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring and can assist with implementing protocols to avoid encountering such circumstances probability is used to design games of chance so that casinos can make a guaranteed profit yet provide payouts to players that are frequent enough to encourage continued play the discovery of rigorous methods to assess and combine probability assessments has changed society another significant application of probability theory in everyday life is reliability many consumer products such as automobiles and consumer electronics use reliability theory in product design to reduce the probability of failure failure probability may influence a manufacturers decisions on a products warranty the cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory mathematical treatment see also probability axioms consider an experiment that can produce a number of results the collection of all possible results is called the sample space of the experiment the power set of the sample space is formed by considering all different collections of possible results for example rolling a dice can produce six possible results one collection of possible results gives an odd number on the dice thus the subset 135 is an element of the power set of the sample space of dice rolls these collections are called events in this case 135 is the event that the dice falls on some odd number if the results that actually occur fall in a given event the event is said to have occurred a probability is a way of assigning every event a value between zero and one with the requirement that the event made up of all possible results in our example the event 123456 is assigned a value of one to qualify as a probability the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events events with no common results eg the events 16 3 and 24 are all mutually exclusive the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events the probability of an event a is written as p a displaystyle pa p a displaystyle pa or pr a displaystyle textpra this mathematical definition of probability can extend to infinite sample spaces and even uncountable sample spaces using the concept of a measure the opposite or complement of an event a is the event that is the event of a not occurring often denoted as a a a displaystyle overline aacomplement neg a or a displaystyle sim a its probability is given by pnot a 1 pa as an example the chance of not rolling a six on a sixsided die is 1 chance of rolling a six 1 1 6 5 6 displaystyle 1tfrac 16tfrac 56 see complementary event for a more complete treatment if two events a and b occur on a single performance of an experiment this is called the intersection or joint probability of a and b denoted as p a b displaystyle pacap b independent events if two events a and b are independent then the joint probability is p a and b p a b p a p b displaystyle pambox and bpacap bpapb for example if two coins are flipped the chance of both being heads is 1 2 1 2 1 4 displaystyle tfrac 12times tfrac 12tfrac 14 mutually exclusive events if either event a or event b but never both occurs on a single performance of an experiment then they are called mutually exclusive events if two events are mutually exclusive then the probability of both occurring is denoted as p a b displaystyle pacap b p a and b p a b 0 displaystyle pambox and bpacap b0 if two events are mutually exclusive then the probability of either occurring is denoted as p a b displaystyle pacup b p a or b p a b p a p b p a b p a p b 0 p a p b displaystyle pambox or bpacup bpapbpacap bpapb0papb for example the chance of rolling a 1 or 2 on a sixsided die is p 1 or 2 p 1 p 2 1 6 1 6 1 3 displaystyle p1mbox or 2p1p2tfrac 16tfrac 16tfrac 13 not mutually exclusive events if the events are not mutually exclusive then p a or b p a b p a p b p a and b displaystyle pleftahbox or brightpacup bpleftarightpleftbrightpleftambox and bright 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 jqk or one that is both is 13 52 12 52 3 52 11 26 displaystyle tfrac 1352tfrac 1252tfrac 352tfrac 1126 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 conditional probability conditional probability is the probability of some event a given the occurrence of some other event b conditional probability is written p a b displaystyle pamid b and is read the probability of a given b it is defined by p a b p a b p b displaystyle pamid bfrac pacap bpb if p b 0 displaystyle pb0 then p a b displaystyle pamid b is formally undefined by this expression however it is possible to define a conditional probability for some zeroprobability events using a algebra of such events such as those arising from a continuous random variable 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 1 2 displaystyle 12 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 1 3 displaystyle 13 since only 1 red and 2 blue balls would have been remaining inverse probability in probability theory and applications bayes rule relates the odds of event a 1 displaystyle a1 to event a 2 displaystyle a2 before prior to and after posterior to conditioning on another event b displaystyle b the odds on a 1 displaystyle a1 to event a 2 displaystyle a2 is simply the ratio of the probabilities of the two events when arbitrarily many events a displaystyle a are of interest not just two the rule can be rephrased as posterior is proportional to prior times likelihood p a b p a p b a displaystyle pabpropto papba where the proportionality symbol means that the left hand side is proportional to ie equals a constant times the right hand side as a displaystyle a varies for fixed or given b displaystyle b lee 2012 bertsch mcgrayne 2012 in this form it goes back to laplace 1774 and to cournot 1843 see fienberg 2005 see inverse probability and bayes rule summary of probabilities summary of probabilities event probability a p a 0 1 displaystyle pain not a p a 1 p a displaystyle pacomplement 1pa a or b p a b p a p b p a b p a b p a p b if a and b are mutually exclusive displaystyle beginalignedpacup bpapbpacap bpacup bpapbqquad mboxif a and b are mutually exclusiveendaligned a and b p a b p a b p b p b a p a p a b p a p b if a and b are independent displaystyle beginalignedpacap bpabpbpbapapacap bpapbqquad mboxif a and b are independentendaligned a given b p a b p a b p b p b a p a p b displaystyle pamid bfrac pacap bpbfrac pbapapb relation to randomness and probability in quantum mechanics randomness see also quantum fluctuation interpretations in a deterministic universe based on newtonian concepts there would be no probability if all conditions were known laplaces demon but there are situations in which sensitivity to initial conditions exceeds our ability to measure them ie know them in the case of a roulette wheel if the force of the hand and the period of that force are known the number on which the ball will stop would be a certainty though as a practical matter this would likely be true only of a roulette wheel that had not been exactly levelled as thomas a bass newtonian casino revealed this also assumes knowledge of inertia and friction of the wheel weight smoothness and roundness of the ball variations in hand speed during the turning and so forth a probabilistic description can thus be more useful than newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of a roulette wheel physicists face the same situation in kinetic theory of gases where the system while deterministic in principle is so complex with the number of molecules typically the order of magnitude of the avogadro constant 70236020000000000006021023 that only a statistical description of its properties is feasible probability theory is required to describe quantum phenomena in some modern interpretations of the statistical mechanics of measurement quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes mathematics portal logic portal outline of probability chance disambiguation class membership probabilities contingency equiprobability heuristics in judgment and decisionmaking probability theory randomness statistics estimators estimation theory probability density function in law balance of probabilities notes bibliography kallenberg o 2005 probabilistic symmetries and invariance principles springer verlag new york 510 ppisbn0387251154 kallenberg o 2002 foundations of modern probability 2nd ed springer series in statistics 650 ppisbn0387953132 olofsson peter 2005 probability statistics and stochastic processes wileyinterscience 504 pp isbn0471679690 wikiquote has quotations related to probability wikibooks has more on the topic of probability wikimedia commons has media related to probability library resources about probability resources in your library virtual laboratories in probability and statistics univ of alahuntsville probability on in our time at the bbc probability and statistics ebook edwin thompson jaynes probability theory the logic of science preprint washington university 1996 html index with links to postscript files and pdf first three chapters people from the history of probability and statistics univ of southampton probability and statistics on the earliest uses pages univ of southampton earliest uses of symbols in probability and statistics on earliest uses of various mathematical symbols a tutorial on probability and bayes theorem devised for firstyear oxford university students pdf file of an anthology of chance operations 1963 at ubuweb introduction to probability ebook by charles grinstead laurie snell source gnu free documentation license in english in italian bruno de finetti probabilit e induzione bologna clueb 1993 isbn8880911767 digital version richard p feynmans lecture on probability vtelogic outline history fields argumentation theory axiology critical thinking logic in computer science mathematical logic metalogic metamathematics nonclassical logic philosophical 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