3620:
2911:
3615:{\displaystyle {\begin{aligned}P\left(A\cup B\cup C\right)=&P\left(\left(A\cup B\right)\cup C\right)\\=&P\left(A\cup B\right)+P\left(C\right)-P\left(\left(A\cup B\right)\cap C\right)\\=&P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)+P\left(C\right)-P\left(\left(A\cap C\right)\cup \left(B\cap C\right)\right)\\=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-\left(P\left(A\cap C\right)+P\left(B\cap C\right)-P\left(\left(A\cap C\right)\cap \left(B\cap C\right)\right)\right)\\P\left(A\cup B\cup C\right)=&P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)\end{aligned}}}
2045:
1088:. 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. The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old."
1098:
7156:
5122:
118:
651:
181:
692:
991:
810:, dealing with probabilities in the context of real experiments). For example, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:
1521:
70:
6796:
5136:
5894:. The earliest mention is contained in a letter to Bateson from Francis Galton dated October 1, 1905 (Edwards 2012). We have the testimony of Bateson (1909, p. 57) that "For the introduction of this system , which greatly simplifies difficult cases, I am indebted to Mr. Punnett." The first published diagrams appeared in 1906. when Punnett published the second edition of his
980:
4653:
5024:
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
960:
shows that there has been an interest in quantifying the ideas of probability throughout history, 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
1462:
An 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 commodity's
1575:
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 {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually
976:, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances." However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.
710:
and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and
4837:
1080:
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
1013:
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
1380:
is a representation of its concepts in formal terms – that 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.
5898:, he used a slightly different format ( Punnett 1907, p. 45) In the third edition (Punnett 1911, p. 34) he reverted to the arrangement with a description of the construction of what he called the "chessboard" method (although in truth it is more like a multiplication table).
5236:
In the context of the book that this is quoted from, it is the theory of probability and the logic behind it that governs the phenomena of such things compared to rash predictions that rely on pure luck or mythological arguments such as gods of luck helping the winner of the
5019:
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'
1562:
of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset {1,3,5} is an element of the
2725:
842:. The degree of belief has been interpreted as "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E", although that interpretation is not universally agreed upon. The most popular version of subjective probability is
1478:). 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
1567:
of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.
2904:
since among 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.
711:"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 1/2 (which could also be written as 0.5 or 50%).
2588:
2823:
3991:
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. For example, if a red ball was taken, then the probability of picking a red ball again would be
2465:
4496:
2902:
4975:
4664:
1576:
exclusive events (events with no common results, such as the events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.
862:, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.
2112:
2600:
802:– in a purely theoretical setting (like tossing a coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. This is referred to as
828:
of an experiment's outcome when the experiment is repeated indefinitely. This interpretation considers probability to be the relative frequency "in the long run" of outcomes. A modification of this is
2039:
3763:
1779:
4669:
4501:
2916:
2245:
5215:
interval , there are an infinite number of possible outcomes, and the probability of any given outcome being observed — for instance, exactly 7 — is 0. This means that an observation will
4485:
1412:, probability is taken as a primitive (i.e., not further analyzed), and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the
1189:
4310:
3947:
1493:
and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's
6853:
2730:
2293:
1673:
1730:
1927:
1463:
prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily rationally. The theory of
3835:
3676:
2288:
2179:
4421:
1804:
6329:
1556:
4133:
3800:
4214:
4187:
4103:
4049:
4021:
3989:
2476:
1642:
1613:
1241:
gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by
Laplace (1810, 1812), Gauss (1823),
768:
to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of
4160:
4648:{\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}}}
4350:
4330:
4234:
3879:
3859:
1231:
1211:
2830:
7011:
7006:
148:
6846:
4832:{\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}}}
4848:
1675:. This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.
6968:
472:
1474:
In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biological
4023:
since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be
846:, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective)
5688:
6839:
5814:
835:
1954:
6998:
3691:
678:
2055:
4312:
where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as
5827:
Singh, Laurie (2010) "Whither
Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.
5702:
5095:
2184:
5227:. Ultimately some specific outcome (with probability 0) will be observed, and one possibility for that specific outcome is exactly 7.
84:
6063:
1482:
so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.
141:
5574:
2908:
This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows:
1233:
is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as
832:, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
7103:
1129:
876:
465:
99:
94:
6820:
7026:
7021:
6150:
5790:
5636:
5611:
5502:
5479:
5449:
5424:
5399:
5320:
5262:
4352:(Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).
1735:
817:
assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is
6326:
2827:
For example, when drawing a card from a deck of cards, the chance of getting a heart or a face card (J, Q, K) (or both) is
614:
6340:
6279:
6943:
6898:
6452:
5472:
The
Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference
134:
2720:{\displaystyle P\left(A{\hbox{ or }}B\right)=P(A\cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).}
7128:
7068:
7041:
6581:
855:
5257:"Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009),
5083:
6993:
6953:
6878:
6730:
6373:
6317:
6209:
6195:
6181:
5982:
5523:
5283:
3949:
However, it is possible to define a conditional probability for some zero-probability events, for example by using a
1945:
1274:
671:
458:
446:
405:
89:
7063:
6981:
6976:
6814:
6294:
5837:
5054:
was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of
4432:
3960:
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
1534:
Consider an experiment that can produce a number of results. The collection of all possible results is called the
6406:
6222:
5208:
787:
336:
272:
7181:
6735:
6349:
847:
384:
245:
5911:
Gao, J.Z.; Fong, D.; Liu, X. (April 2011). "Mathematical analyses of casino rebate systems for VIP gambling".
4243:
1094:(1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
7123:
7113:
6938:
6933:
859:
5364:
3884:
6988:
6948:
1266:
1055:
664:
104:
50:
5548:
1427:, but those are essentially different and not compatible with the usually-understood laws of probability.
7036:
7031:
6928:
6913:
6810:
6308:
6124:
5855:
4994:
1509:
1242:
872:
781:
507:
5390:
Jaynes, E.T. (2003). "Section A.2 The de
Finetti system of probability". In Bretthorst, G. Larry (ed.).
7146:
7088:
7058:
6761:
6255:
6142:
3954:
1580:
1505:
1420:
1401:
707:
240:
5295:
6740:
6665:
6447:
5063:
2248:
2139:
2044:
629:
619:
609:
356:
1647:
6923:
6893:
6638:
5155:
1486:
1452:
814:
415:
410:
299:
284:
17:
5440:
Jaynes, E.T. (2003). "Section 5.3 Converging and diverging views". In
Bretthorst, G. Larry (ed.).
7078:
7053:
6888:
6883:
6722:
5886:
Punnett's square seems to have been a development of 1905, too late for the first edition of his
5026:
4136:
3838:
3631:
1952:
1894:
1250:
1109:
1042:
818:
799:
749:
394:
265:
3805:
3646:
2258:
2149:
7108:
7093:
5170:
5149:
5067:
1572:
945:
829:
823:
624:
599:
502:
289:
79:
6309:
A tutorial on probability and Bayes' theorem devised for first-year Oxford
University students
5110:
is invoked to account for the appearance of subjectively probabilistic experimental outcomes.
4379:
1784:
1697:
6963:
6903:
6776:
6289:
6271:
2583:{\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.}
1541:
1405:
1389:
1336:
1270:
1105:
951:
933:
839:
807:
634:
548:
430:
389:
294:
260:
5710:
4108:
3770:
1451:
to determine pricing and make trading decisions. Governments apply probabilistic methods in
1097:
7176:
7133:
7118:
7083:
7073:
6918:
6695:
6541:
5755:
5336:
Finetti, Bruno de (1970). "Logical foundations and measurement of subjective probability".
5050:
is required to describe quantum phenomena. A revolutionary discovery of early 20th century
4192:
4165:
4081:
2818:{\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)}
1501:
1456:
1238:
1085:
843:
594:
553:
522:
420:
314:
207:
109:
5091:
4026:
3995:
3963:
1618:
1589:
8:
7098:
7048:
6958:
6908:
6650:
6633:
6613:
6576:
6525:
6520:
6462:
6399:
5107:
5004:
4216:
is simply the ratio of the probabilities of the two events. When arbitrarily many events
4059:
2135:
can occur but never both simultaneously, then they are called mutually exclusive events.
1865:
1258:
929:
851:
379:
321:
309:
304:
6831:
5842:"Reginald Crundall Punnett: First Arthur Balfour Professor of Genetics, Cambridge, 1912"
4142:
2460:{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)}
1061:
for histories of the early development of the very concept of mathematical probability.
7160:
6586:
6515:
6472:
6101:
5928:
5872:
5846:
5841:
5808:
5604:
Quantum leap : from Dirac and
Feynman, across the universe, to human body and mind
5580:
5127:
5047:
5008:
4335:
4315:
4219:
4065:
3864:
3844:
2897:{\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}},}
2122:
1880:
1529:
1464:
1444:
1424:
1413:
1377:
1367:
1332:
1313:
1262:
1216:
1196:
1023:
994:
925:
720:
715:
654:
558:
366:
255:
195:
172:
122:
31:
6087:
Riedi, P.C. (1976). Kinetic Theory of Gases-I. In: Thermal
Physics. Palgrave, London.
5033:, is so complex (with the number of molecules typically the order of magnitude of the
1384:
There have been at least two successful attempts to formalize probability, namely the
7155:
6800:
6771:
6766:
6756:
6690:
6618:
6503:
6369:
6205:
6191:
6177:
6146:
5978:
5932:
5877:
5796:
5786:
5670:
5653:
5632:
5607:
5519:
5498:
5475:
5445:
5420:
5395:
5349:
5316:
5279:
5258:
5141:
5121:
5055:
5034:
4070:
1448:
1440:
1397:
1393:
1355:
1340:
1293:
1037:
650:
425:
331:
230:
117:
6705:
6431:
6426:
6359:
5955:
5920:
5867:
5859:
5733:
5665:
5345:
1879:
occur on a single performance of an experiment, this is called the intersection or
1409:
1301:
1297:
1246:
1091:
1065:
1015:
1006:
983:
757:
753:
250:
180:
6236:
4970:{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,}
3622:
It can be seen, then, that this pattern can be repeated for any number of events.
6551:
6493:
6344:
6333:
6266:
6169:
6002:
5924:
5160:
5079:
5075:
1479:
1475:
1305:
1032:
1027:
965:
769:
576:
326:
277:
6379:
5863:
5551:
Some laws and problems in classical probability and how Cardano anticipated them
901:
6805:
6498:
6477:
6392:
6337:
5490:
1351:
1282:
1254:
1077:
990:
725:
604:
341:
6088:
5106:. In some modern interpretations of the statistical mechanics of measurement,
5025:
repeated rolls of a roulette wheel. Physicists face the same situation in the
4982:
2052:
For example, if two coins are flipped, then the chance of both being heads is
1124:, editor of "The Analyst" (1808), first deduced the law of facility of error,
7170:
6655:
6596:
6313:
6247:
Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)
6100:
Burgin, Mark (2010). "Interpretations of Negative Probabilities". p. 1.
5800:
5197:
5190:
5066:, it deals with probabilities of observing, the outcome being explained by a
5059:
1328:
1324:
1234:
1121:
1019:
854:. The product of the prior and the likelihood, when normalized, results in a
529:
214:
6295:
Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
6290:
People from the History of Probability and Statistics (Univ. of Southampton)
6251:
5707:
StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies
968:, "Before the middle of the seventeenth century, the term 'probable' (Latin
956:
The scientific study of probability is a modern development of mathematics.
6645:
6467:
6283:
5959:
5881:
5204:
5012:
1535:
1317:
1309:
441:
351:
235:
39:
3950:
1485:
Another significant application of probability theory in everyday life is
691:
6867:
6680:
6628:
6303:
5467:
5308:
5212:
5165:
5099:
5071:
5000:
1490:
1069:
1047:
961:
of probability, fundamental issues are still obscured by superstitions.
761:
733:
703:
493:
361:
202:
35:
5015:
exceeds our ability to measure them, i.e. know them). In the case of a
6623:
6591:
6556:
4988:
1825:. As an example, the chance of not rolling a six on a six-sided die is
1468:
1385:
1335:
theory and its applications. The modern theory of probability based on
913:
882:
795:
791:
765:
729:
572:
562:
219:
165:
43:
6299:
5007:
concepts, there would be no probability if all conditions were known (
2107:{\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}.}
1520:
1120:). In ignorance of Legendre's contribution, an Irish-American writer,
6685:
6546:
6457:
5044:) that only a statistical description of its properties is feasible.
5021:
2048:
Events A and B depicted as independent vs non-independent in space Ω.
1564:
1559:
905:
891:
6106:
1084:
The first two laws of error that were proposed both originated with
1026:(1657) gave the earliest known scientific treatment of the subject.
6606:
5087:
5016:
1494:
1002:
957:
917:
745:
544:
534:
346:
6364:
5536:
The Science of Conjecture: Evidence and Probability Before Pascal,
1288:
In the nineteenth century, authors on the general theory included
821:, which claims that the probability of a random event denotes the
6863:
6670:
6601:
5629:
The Science of Conjecture: Evidence and Probability Before Pascal
5103:
5051:
1419:
There are other methods for quantifying uncertainty, such as the
1289:
909:
741:
737:
539:
6246:
5196:
takes place, whereas a probability of 1 indicates than an event
1114:
Nouvelles méthodes pour la détermination des orbites des comètes
6508:
6321:
5090:: "I am convinced that God does not play dice". Like Einstein,
1373:
997:
published one of the first books on probability (17th century).
5189:
Strictly speaking, a probability of 0 indicates that an event
2592:
6700:
6415:
5783:
Understanding and applying basic statistical methods using R
4236:
are of interest, not just two, the rule can be rephrased as
1350:
included Miller, Crofton, McColl, Wolstenholme, Watson, and
6660:
5606:. Singapore; Hackensack, NJ: World Scientific. p. 16.
4983:
Relation to randomness and probability in quantum mechanics
4075:
2597:
If the events are not (necessarily) mutually exclusive then
2470:
2469:
For example, the chance of rolling a 1 or 2 on a six-sided
1436:
1046:(1718) treated the subject as a branch of mathematics. See
1010:
920:. In a sense, this differs much from the modern meaning of
695:
The probabilities of rolling several numbers using two dice
5892:
Report III to the Evolution Committee of the Royal Society
5276:
An Introduction to Probability Theory and Its Applications
979:
6861:
6261:
6121:
Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt.
6064:"Conditional probability with respect to a sigma-algebra"
5415:
Hogg, Robert V.; Craig, Allen; McKean, Joseph W. (2004).
1512:
are also examples of applications of probability theory.
1213:
is a constant depending on precision of observation, and
838:
assign numbers per subjective probability, that is, as a
6384:
2034:{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).}
858:
that incorporates all the information known to date. By
5977:(8th ed.). Pearson Prentice Hall. pp. 26–27.
5203:
takes place. This is an important distinction when the
3758:{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}\,}
1774:{\displaystyle {\overline {A}},A^{\complement },\neg A}
27:
Branch of mathematics concerning chance and uncertainty
6300:
Earliest Uses of Symbols in Probability and Statistics
5363:
Hájek, Alan (21 October 2002). Edward N. Zalta (ed.).
4819:
4635:
2880:
2865:
2850:
2835:
2700:
2616:
2566:
2551:
2536:
2490:
2307:
2198:
2090:
2075:
2060:
1968:
7144:
5602:
Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008).
4851:
4667:
4499:
4435:
4382:
4338:
4318:
4246:
4222:
4195:
4168:
4145:
4111:
4084:
4029:
3998:
3966:
3887:
3867:
3847:
3808:
3773:
3694:
3649:
2914:
2833:
2733:
2603:
2479:
2296:
2261:
2187:
2152:
2058:
1957:
1897:
1787:
1738:
1700:
1650:
1621:
1592:
1544:
1219:
1199:
1132:
1081:
continuous errors and describes a probability curve.
6190:
2nd ed. Springer Series in Statistics. 650 pp.
5117:
5687:Wilson EB (1923) "First and second laws of error".
5098:the wave function, believed quantum mechanics is a
4238:
posterior is proportional to prior times likelihood
1392:formulation. In Kolmogorov's formulation (see also
916:in Europe, and often correlated with the witness's
6174:Probabilistic Symmetries and Invariance Principles
5946:Gorman, Michael F. (2010). "Management Insights".
5601:
5538:Johns Hopkins University Press. (pp. 22, 113, 127)
5251:
5070:when an observation is made. However, the loss of
4969:
4831:
4647:
4479:
4415:
4344:
4324:
4304:
4228:
4208:
4181:
4154:
4127:
4097:
4043:
4015:
3983:
3941:
3873:
3853:
3829:
3794:
3757:
3670:
3614:
2896:
2817:
2719:
2582:
2459:
2282:
2239:
2173:
2106:
2033:
1921:
1798:
1773:
1724:
1667:
1636:
1607:
1550:
1471:on pricing, on policy, and on peace and conflict.
1435:Probability theory is applied in everyday life in
1225:
1205:
1183:
924:, which in contrast is a measure of the weight of
6202:Probability, Statistics, and Stochastic Processes
7168:
5854:(1). Gonville and Caius College, Cambridge, UK:
5518:Cambridge University Press. (1992). pp. 54–55 .
5062:evolves deterministically but, according to the
2240:{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0}
1118:New Methods for Determining the Orbits of Comets
5780:
5689:Journal of the American Statistical Association
5414:
5830:
6847:
6400:
6304:Earliest Uses of Various Mathematical Symbols
6278:. Preprint: Washington University, (1996). –
5683:
5681:
5419:(6th ed.). Upper Saddle River: Pearson.
5102:approximation of an underlying deterministic
3953:of such events (such as those arising from a
1076:(posthumous, 1722), but a memoir prepared by
672:
466:
142:
5463:
5461:
4480:{\displaystyle P(A^{\complement })=1-P(A)\,}
2116:
1416:are the same, except for technical details.
6089:https://doi.org/10.1007/978-1-349-15669-6_8
4355:
4135:before (prior to) and after (posterior to)
3639:, given the occurrence of some other event
2593:Not (necessarily) mutually exclusive events
895:
6854:
6840:
6407:
6393:
6176:. Springer-Verlag, New York. 510 pp.
5910:
5813:: CS1 maint: location missing publisher (
5678:
5444:(1 ed.). Cambridge University Press.
5394:(1 ed.). Cambridge University Press.
4995:Quantum fluctuation § Interpretations
3625:
679:
665:
473:
459:
149:
135:
68:
6380:Richard Feynman's Lecture on probability.
6280:HTML index with links to PostScript files
6123:Letter to Max Born, 4 December 1926, in:
6105:
5871:
5669:
5651:
5458:
4966:
4476:
3754:
1524:Calculation of probability (risk) vs odds
1515:
1184:{\displaystyle \phi (x)=ce^{-h^{2}x^{2}}}
6276:Probability Theory: The Logic of Science
5654:"Thomas Simpson and the arithmetic mean"
5626:
5528:
5442:Probability Theory: The Logic of Science
5392:Probability Theory: The Logic of Science
5029:, where the system, while deterministic
4305:{\displaystyle P(A|B)\propto P(A)P(B|A)}
2043:
1864:For a more comprehensive treatment, see
1538:of the experiment, sometimes denoted as
1519:
1285:of a single observation, is well known.
1096:
989:
978:
690:
5836:
5753:
5508:
5417:Introduction to Mathematical Statistics
5369:The Stanford Encyclopedia of Philosophy
5335:
5307:
5278:, vol. 1, 3rd ed., (1968), Wiley,
1467:emerged to describe the effect of such
1346:On the geometric side, contributors to
1320:improved the exposition of the theory.
873:History of probability § Etymology
14:
7169:
6969:Electrical and electronics engineering
6954:Developmental and reproductive biology
6099:
5945:
5439:
5389:
5078:did not meet with universal approval.
4053:
3942:{\displaystyle P(A\cap B)=P(A)P(B)=0.}
1009:demonstrated the efficacy of defining
877:Glossary of probability and statistics
6835:
6388:
6336:, by Charles Grinstead, Laurie Snell
6136:
6000:
5996:
5994:
5703:""Adrien-Marie Legendre" (version 9)"
5362:
5268:
5011:) (but there are situations in which
3643:. Conditional probability is written
1951:
1931:
1690:is the event (that is, the event of
6125:Einstein/Born Briefwechsel 1916–1955
5972:
5516:Probability and the Art of Judgment,
1331:, which played an important role in
6582:Analytic and synthetic propositions
6453:Formal semantics (natural language)
6327:Introduction to Probability – eBook
5890:(May 1905) but much in evidence in
850:. These data are incorporated in a
24:
6314:U B U W E B :: La Monte Young
6188:Foundations of Modern Probability,
5991:
5700:
5652:Shoesmith, Eddie (November 1985).
5631:. Johns Hopkins University Press.
5572:
5313:The Logic of Statistical Inference
5207:is infinite. For example, for the
3678:, and is read "the probability of
1765:
1545:
1489:. Many consumer products, such as
856:posterior probability distribution
775:
714:These concepts have been given an
25:
7193:
6999:Genetics and evolutionary biology
6318:An Anthology of Chance Operations
6216:
5838:Edwards, Anthony William Fairbank
5731:
5013:sensitivity to initial conditions
4637:if A and B are mutually exclusive
3841:by this expression. In this case
3635:is the probability of some event
1694:not occurring), often denoted as
7154:
6794:
6267:Probability and Statistics EBook
5365:"Interpretations of Probability"
5134:
5120:
1257:(1870). Other contributors were
649:
179:
116:
7003:Cellular and molecular biology
6163:
6130:
6114:
6093:
6081:
6056:
6047:
6038:
6029:
6020:
5966:
5939:
5904:
5821:
5774:
5756:"Andrei Nikolaevich Kolmogorov"
5747:
5725:
5694:
5645:
5620:
5595:
5579:. Second Moment. Archived from
5566:
5541:
5484:
5433:
5230:
5219:be exactly 7. However, it does
5209:continuous uniform distribution
4817:
4633:
1827:1 – (chance of rolling a six) =
1430:
6350:GNU Free Documentation License
5913:International Gambling Studies
5576:A Brief History of Probability
5474:, Cambridge University Press,
5408:
5383:
5356:
5329:
5315:. Cambridge University Press.
5301:
5289:
5183:
4960:
4954:
4946:
4940:
4934:
4927:
4920:
4905:
4899:
4891:
4879:
4867:
4855:
4814:
4808:
4802:
4796:
4783:
4771:
4761:
4755:
4749:
4742:
4735:
4726:
4720:
4714:
4707:
4700:
4687:
4675:
4630:
4624:
4615:
4609:
4596:
4584:
4574:
4562:
4553:
4547:
4538:
4532:
4519:
4507:
4473:
4467:
4452:
4439:
4410:
4398:
4392:
4386:
4299:
4292:
4285:
4279:
4273:
4264:
4257:
4250:
3930:
3924:
3918:
3912:
3903:
3891:
3824:
3812:
3783:
3777:
3748:
3742:
3734:
3722:
3710:
3698:
3665:
3653:
2648:
2636:
2529:
2523:
2514:
2508:
2499:
2483:
2454:
2448:
2439:
2433:
2418:
2412:
2403:
2397:
2388:
2376:
2367:
2361:
2352:
2346:
2337:
2325:
2316:
2300:
2277:
2265:
2228:
2216:
2207:
2191:
2168:
2156:
2025:
2019:
2013:
2007:
1998:
1986:
1977:
1961:
1948:then the joint probability is
1913:
1901:
1806:; its probability is given by
1668:{\displaystyle {\text{Pr}}(A)}
1662:
1656:
1631:
1625:
1602:
1596:
1142:
1136:
848:prior probability distribution
718:mathematical formalization in
246:Collectively exhaustive events
13:
1:
6204:, Wiley-Interscience. 504 pp
6139:Schrödinger: Life and Thought
5975:A First course in Probability
5244:
1443:. The insurance industry and
5925:10.1080/14459795.2011.552575
5671:10.1016/0315-0860(85)90044-8
5350:10.1016/0001-6918(70)90012-0
1744:
1455:, entitlement analysis, and
1245:(1825, 1826), Hagen (1837),
1052:The Emergence of Probability
866:
51:Probability (disambiguation)
7:
5864:10.1534/genetics.112.143552
5856:Genetics Society of America
5754:Vitanyi, Paul M.B. (1988).
5495:Introduction to Probability
5113:
4332:varies, for fixed or given
1922:{\displaystyle P(A\cap B).}
1510:natural language processing
1506:statistical language models
1308:(1873), Liagre, Didion and
1112:, and introduced it in his
782:Probability interpretations
10:
7198:
7104:Probability and statistics
6143:Cambridge University Press
5743:. University of Cambridge.
5147:
4992:
4986:
4821:if A and B are independent
4057:
3955:continuous random variable
3830:{\displaystyle P(A\mid B)}
3671:{\displaystyle P(A\mid B)}
2283:{\displaystyle P(A\cup B)}
2251:, then the probability of
2174:{\displaystyle P(A\cap B)}
2142:, then the probability of
2120:
1527:
1365:
949:
943:
939:
880:
870:
860:Aumann's agreement theorem
779:
724:, which is used widely in
589:Relationship with sciences
48:
29:
6874:
6789:
6749:
6721:
6714:
6666:Necessity and sufficiency
6569:
6534:
6486:
6440:
6422:
6414:
6237:Resources in your library
5973:Ross, Sheldon M. (2010).
5298:. The Britannica website.
5064:Copenhagen interpretation
4842:
4658:
4490:
4426:
4373:
4368:
4365:
4361:Summary of probabilities
2117:Mutually exclusive events
1361:
1327:introduced the notion of
1059:The Science of Conjecture
928:, and is arrived at from
6368:, Bologna, CLUEB, 1993.
6357:(in English and Italian)
5781:Wilcox, Rand R. (2016).
5701:Seneta, Eugene William.
5627:Franklin, James (2001).
5176:
4416:{\displaystyle P(A)\in }
4356:Summary of probabilities
2255:occurring is denoted as
2146:occurring is denoted as
1799:{\displaystyle {\sim }A}
1725:{\displaystyle A',A^{c}}
1453:environmental regulation
1304:(1860), Helmert (1872),
1296:(1816), Littrow (1833),
416:Law of total probability
411:Conditional independence
300:Exponential distribution
285:Probability distribution
30:Not to be confused with
6899:Artificial intelligence
6365:Probabilità e induzione
6200:Olofsson, Peter (2005)
5785:. Hoboken, New Jersey.
5223:mean that exactly 7 is
5027:kinetic theory of gases
3881:are independent, since
3632:Conditional probability
3626:Conditional probability
1551:{\displaystyle \Omega }
1408:on a class of sets. In
1110:method of least squares
1036:(posthumous, 1713) and
900:, which can also mean "
819:frequentist probability
804:theoretical probability
750:artificial intelligence
395:Conditional probability
7129:Structural engineering
7069:Mechanical engineering
7042:Western and South Asia
6286:(first three chapters)
6186:Kallenberg, O. (2002)
6053:Olofsson (2005) p. 29.
6044:Olofsson (2005) p. 35.
5960:10.1287/mnsc.1090.1132
5741:Statistical Laboratory
5371:(Winter 2012 ed.)
5171:Heuristic (psychology)
5150:Outline of probability
5068:wave function collapse
4971:
4833:
4649:
4481:
4417:
4346:
4326:
4306:
4230:
4210:
4183:
4156:
4129:
4128:{\displaystyle A_{2},}
4099:
4045:
4017:
3985:
3943:
3875:
3855:
3831:
3796:
3795:{\displaystyle P(B)=0}
3759:
3672:
3616:
2898:
2819:
2721:
2584:
2461:
2284:
2241:
2175:
2108:
2049:
2035:
1923:
1800:
1775:
1726:
1669:
1638:
1609:
1579:The probability of an
1552:
1525:
1516:Mathematical treatment
1421:Dempster–Shafer theory
1358:for more information.
1277:'s (1856) formula for
1227:
1207:
1185:
1102:
1068:may be traced back to
1001:The sixteenth-century
998:
987:
946:History of probability
896:
830:propensity probability
696:
337:Continuous or discrete
290:Bernoulli distribution
123:Mathematics portal
7182:Dimensionless numbers
6994:Environmental science
6879:Aerospace engineering
6801:Philosophy portal
6343:25 March 2012 at the
6272:Edwin Thompson Jaynes
6035:Olofsson (2005), p. 9
6026:Olofsson (2005) p. 8.
6007:mathworld.wolfram.com
4972:
4834:
4650:
4482:
4418:
4347:
4327:
4307:
4231:
4211:
4209:{\displaystyle A_{2}}
4184:
4182:{\displaystyle A_{1}}
4157:
4130:
4100:
4098:{\displaystyle A_{1}}
4046:
4018:
3986:
3944:
3876:
3856:
3832:
3797:
3760:
3673:
3617:
2899:
2820:
2722:
2585:
2462:
2285:
2242:
2176:
2109:
2047:
2036:
1924:
1801:
1776:
1727:
1670:
1639:
1610:
1553:
1523:
1404:and probability as a
1378:theory of probability
1348:The Educational Times
1271:Giovanni Schiaparelli
1228:
1208:
1186:
1108:(1805) developed the
1106:Adrien-Marie Legendre
1100:
993:
982:
952:History of statistics
950:Further information:
934:statistical inference
881:Further information:
808:empirical probability
694:
295:Binomial distribution
6332:27 July 2011 at the
6137:Moore, W.J. (1992).
5658:Historia Mathematica
5534:Franklin, J. (2001)
4849:
4665:
4497:
4433:
4380:
4336:
4316:
4244:
4220:
4193:
4166:
4143:
4109:
4082:
4044:{\displaystyle 2/3.}
4027:
4016:{\displaystyle 1/3,}
3996:
3984:{\displaystyle 1/2;}
3964:
3885:
3865:
3845:
3806:
3771:
3692:
3686:". It is defined by
3647:
2912:
2831:
2731:
2601:
2477:
2294:
2259:
2185:
2150:
2056:
1955:
1895:
1785:
1736:
1698:
1648:
1637:{\displaystyle p(A)}
1619:
1608:{\displaystyle P(A)}
1590:
1542:
1502:cache language model
1457:financial regulation
1388:formulation and the
1333:stochastic processes
1217:
1197:
1130:
1101:Carl Friedrich Gauss
1086:Pierre-Simon Laplace
904:", a measure of the
844:Bayesian probability
554:Discrete mathematics
421:Law of large numbers
390:Marginal probability
315:Poisson distribution
164:Part of a series on
49:For other uses, see
6463:Philosophy of logic
6001:Weisstein, Eric W.
5108:quantum decoherence
5003:universe, based on
4362:
4060:Inverse probability
4054:Inverse probability
1866:Complementary event
1571:A probability is a
1414:laws of probability
1400:are interpreted as
1043:Doctrine of Chances
930:inductive reasoning
852:likelihood function
488:Part of a series on
380:Complementary event
322:Probability measure
310:Pareto distribution
305:Normal distribution
85:Catalog of articles
6762:Rules of inference
6731:Mathematical logic
6473:Semantics of logic
5948:Management Science
5840:(September 2012).
5713:on 3 February 2016
5296:Probability Theory
5128:Mathematics portal
5048:Probability theory
4967:
4829:
4827:
4823:
4645:
4643:
4639:
4477:
4413:
4360:
4342:
4322:
4302:
4226:
4206:
4179:
4155:{\displaystyle B.}
4152:
4125:
4095:
4068:and applications,
4066:probability theory
4041:
4013:
3981:
3939:
3871:
3851:
3827:
3792:
3755:
3668:
3612:
3610:
2894:
2889:
2874:
2859:
2844:
2815:
2717:
2704:
2620:
2580:
2575:
2560:
2545:
2494:
2457:
2311:
2280:
2249:mutually exclusive
2247:If two events are
2237:
2202:
2171:
2140:mutually exclusive
2138:If two events are
2123:Mutual exclusivity
2104:
2099:
2084:
2069:
2050:
2031:
1972:
1932:Independent events
1919:
1796:
1771:
1722:
1665:
1634:
1605:
1548:
1530:Probability axioms
1526:
1465:behavioral finance
1425:possibility theory
1368:Probability theory
1314:Augustus De Morgan
1253:(1844, 1856), and
1223:
1203:
1181:
1103:
1024:Christiaan Huygens
999:
995:Christiaan Huygens
988:
926:empirical evidence
824:relative frequency
788:random experiments
786:When dealing with
721:probability theory
697:
655:Mathematics Portal
431:Boole's inequality
367:Stochastic process
256:Mutual exclusivity
173:Probability theory
32:probability theory
7142:
7141:
7124:Scientific naming
7114:Quantum computing
6944:Computer hardware
6939:Clinical research
6934:Civil engineering
6829:
6828:
6785:
6784:
6619:Deductive closure
6565:
6564:
6504:Critical thinking
6376:(digital version)
6223:Library resources
6152:978-0-521-43767-7
5792:978-1-119-06140-3
5638:978-0-8018-6569-5
5613:978-981-281-927-7
5573:Abrams, William.
5503:978-0-8221-0078-2
5480:978-0-521-68557-3
5451:978-0-521-59271-0
5426:978-0-13-008507-8
5401:978-0-521-59271-0
5338:Acta Psychologica
5322:978-0-521-05165-1
5263:978-0-534-24312-8
5217:almost surely not
5142:Philosophy portal
5092:Erwin Schrödinger
5056:quantum mechanics
5035:Avogadro constant
4980:
4979:
4964:
4909:
4822:
4638:
4345:{\displaystyle B}
4325:{\displaystyle A}
4229:{\displaystyle A}
4139:on another event
3874:{\displaystyle B}
3854:{\displaystyle A}
3752:
2888:
2873:
2858:
2843:
2703:
2619:
2574:
2559:
2544:
2493:
2310:
2201:
2098:
2083:
2068:
1971:
1881:joint probability
1747:
1654:
1508:that are used in
1449:actuarial science
1394:probability space
1356:integral geometry
1341:Andrey Kolmogorov
1339:was developed by
1294:Sylvestre Lacroix
1226:{\displaystyle c}
1206:{\displaystyle h}
1074:Opera Miscellanea
1038:Abraham de Moivre
702:is the branch of
689:
688:
644:
643:
483:
482:
385:Joint probability
332:Bernoulli process
231:Probability space
159:
158:
16:(Redirected from
7189:
7159:
7158:
7150:
6949:Computer science
6856:
6849:
6842:
6833:
6832:
6799:
6798:
6797:
6719:
6718:
6484:
6483:
6448:Computer science
6409:
6402:
6395:
6386:
6385:
6360:Bruno de Finetti
6358:
6157:
6156:
6134:
6128:
6118:
6112:
6111:
6109:
6097:
6091:
6085:
6079:
6078:
6076:
6074:
6068:www.statlect.com
6060:
6054:
6051:
6045:
6042:
6036:
6033:
6027:
6024:
6018:
6017:
6015:
6013:
5998:
5989:
5988:
5970:
5964:
5963:
5943:
5937:
5936:
5908:
5902:
5900:
5884:. pp. 5–6:
5875:
5844:. Perspectives.
5834:
5828:
5825:
5819:
5818:
5812:
5804:
5778:
5772:
5771:
5769:
5767:
5751:
5745:
5744:
5738:
5732:Weber, Richard.
5729:
5723:
5722:
5720:
5718:
5709:. Archived from
5698:
5692:
5685:
5676:
5675:
5673:
5649:
5643:
5642:
5624:
5618:
5617:
5599:
5593:
5592:
5590:
5588:
5570:
5564:
5563:
5561:
5545:
5539:
5532:
5526:
5512:
5506:
5488:
5482:
5465:
5456:
5455:
5437:
5431:
5430:
5412:
5406:
5405:
5387:
5381:
5380:
5378:
5376:
5360:
5354:
5353:
5333:
5327:
5326:
5305:
5299:
5293:
5287:
5274:William Feller,
5272:
5266:
5255:
5238:
5234:
5228:
5187:
5144:
5139:
5138:
5137:
5130:
5125:
5124:
5074:for the sake of
5058:. The objective
5043:
5041:
5022:Newtonian Casino
4976:
4974:
4973:
4968:
4965:
4963:
4949:
4930:
4915:
4910:
4908:
4894:
4874:
4838:
4836:
4835:
4830:
4828:
4824:
4820:
4745:
4710:
4654:
4652:
4651:
4646:
4644:
4640:
4636:
4486:
4484:
4483:
4478:
4451:
4450:
4422:
4420:
4419:
4414:
4363:
4359:
4351:
4349:
4348:
4343:
4331:
4329:
4328:
4323:
4311:
4309:
4308:
4303:
4295:
4260:
4235:
4233:
4232:
4227:
4215:
4213:
4212:
4207:
4205:
4204:
4188:
4186:
4185:
4180:
4178:
4177:
4161:
4159:
4158:
4153:
4134:
4132:
4131:
4126:
4121:
4120:
4104:
4102:
4101:
4096:
4094:
4093:
4050:
4048:
4047:
4042:
4037:
4022:
4020:
4019:
4014:
4006:
3990:
3988:
3987:
3982:
3974:
3948:
3946:
3945:
3940:
3880:
3878:
3877:
3872:
3860:
3858:
3857:
3852:
3836:
3834:
3833:
3828:
3801:
3799:
3798:
3793:
3764:
3762:
3761:
3756:
3753:
3751:
3737:
3717:
3677:
3675:
3674:
3669:
3621:
3619:
3618:
3613:
3611:
3607:
3603:
3576:
3572:
3551:
3547:
3526:
3522:
3501:
3484:
3467:
3448:
3444:
3416:
3412:
3411:
3407:
3406:
3402:
3384:
3380:
3354:
3350:
3329:
3325:
3299:
3295:
3274:
3257:
3240:
3217:
3213:
3212:
3208:
3190:
3186:
3160:
3143:
3139:
3118:
3101:
3078:
3074:
3067:
3063:
3037:
3020:
3016:
2989:
2985:
2978:
2974:
2946:
2942:
2903:
2901:
2900:
2895:
2890:
2881:
2875:
2866:
2860:
2851:
2845:
2836:
2824:
2822:
2821:
2816:
2814:
2810:
2789:
2772:
2755:
2751:
2726:
2724:
2723:
2718:
2713:
2709:
2705:
2701:
2684:
2667:
2629:
2625:
2621:
2617:
2589:
2587:
2586:
2581:
2576:
2567:
2561:
2552:
2546:
2537:
2495:
2491:
2466:
2464:
2463:
2458:
2312:
2308:
2289:
2287:
2286:
2281:
2246:
2244:
2243:
2238:
2203:
2199:
2180:
2178:
2177:
2172:
2127:If either event
2113:
2111:
2110:
2105:
2100:
2091:
2085:
2076:
2070:
2061:
2041:
2038:
2037:
2032:
1973:
1969:
1928:
1926:
1925:
1920:
1863:
1861:
1859:
1858:
1855:
1852:
1845:
1843:
1842:
1839:
1836:
1828:
1824:
1805:
1803:
1802:
1797:
1792:
1780:
1778:
1777:
1772:
1761:
1760:
1748:
1740:
1731:
1729:
1728:
1723:
1721:
1720:
1708:
1674:
1672:
1671:
1666:
1655:
1652:
1643:
1641:
1640:
1635:
1614:
1612:
1611:
1606:
1573:way of assigning
1557:
1555:
1554:
1549:
1302:Richard Dedekind
1298:Adolphe Quetelet
1247:Friedrich Bessel
1232:
1230:
1229:
1224:
1212:
1210:
1209:
1204:
1190:
1188:
1187:
1182:
1180:
1179:
1178:
1177:
1168:
1167:
1092:Daniel Bernoulli
1066:theory of errors
1056:James Franklin's
1016:Pierre de Fermat
1007:Gerolamo Cardano
984:Gerolamo Cardano
899:
840:degree of belief
806:(in contrast to
758:computer science
754:machine learning
681:
674:
667:
653:
517:
516:
485:
484:
475:
468:
461:
251:Elementary event
183:
161:
160:
151:
144:
137:
121:
120:
72:
71:
58:
57:
21:
7197:
7196:
7192:
7191:
7190:
7188:
7187:
7186:
7167:
7166:
7165:
7153:
7145:
7143:
7138:
7037:Hebrew toponyms
7032:Arabic toponyms
6870:
6860:
6830:
6825:
6795:
6793:
6781:
6745:
6736:Boolean algebra
6710:
6561:
6552:Metamathematics
6530:
6482:
6436:
6418:
6413:
6356:
6345:Wayback Machine
6334:Wayback Machine
6243:
6242:
6241:
6231:
6230:
6226:
6219:
6166:
6161:
6160:
6153:
6145:. p. 479.
6135:
6131:
6119:
6115:
6098:
6094:
6086:
6082:
6072:
6070:
6062:
6061:
6057:
6052:
6048:
6043:
6039:
6034:
6030:
6025:
6021:
6011:
6009:
5999:
5992:
5985:
5971:
5967:
5944:
5940:
5909:
5905:
5835:
5831:
5826:
5822:
5806:
5805:
5793:
5779:
5775:
5765:
5763:
5752:
5748:
5736:
5734:"Markov Chains"
5730:
5726:
5716:
5714:
5699:
5695:
5686:
5679:
5650:
5646:
5639:
5625:
5621:
5614:
5600:
5596:
5586:
5584:
5583:on 24 July 2017
5571:
5567:
5559:
5547:
5546:
5542:
5533:
5529:
5514:Jeffrey, R.C.,
5513:
5509:
5489:
5485:
5466:
5459:
5452:
5438:
5434:
5427:
5413:
5409:
5402:
5388:
5384:
5374:
5372:
5361:
5357:
5334:
5330:
5323:
5306:
5302:
5294:
5290:
5273:
5269:
5256:
5252:
5247:
5242:
5241:
5235:
5231:
5188:
5184:
5179:
5161:Equiprobability
5152:
5140:
5135:
5133:
5126:
5119:
5116:
5086:in a letter to
5080:Albert Einstein
5076:instrumentalism
5039:
5037:
5009:Laplace's demon
4997:
4991:
4985:
4950:
4926:
4916:
4914:
4895:
4875:
4873:
4850:
4847:
4846:
4826:
4825:
4818:
4786:
4765:
4764:
4741:
4706:
4690:
4668:
4666:
4663:
4662:
4642:
4641:
4634:
4599:
4578:
4577:
4522:
4500:
4498:
4495:
4494:
4446:
4442:
4434:
4431:
4430:
4381:
4378:
4377:
4358:
4337:
4334:
4333:
4317:
4314:
4313:
4291:
4256:
4245:
4242:
4241:
4221:
4218:
4217:
4200:
4196:
4194:
4191:
4190:
4173:
4169:
4167:
4164:
4163:
4144:
4141:
4140:
4116:
4112:
4110:
4107:
4106:
4089:
4085:
4083:
4080:
4079:
4062:
4056:
4033:
4028:
4025:
4024:
4002:
3997:
3994:
3993:
3970:
3965:
3962:
3961:
3886:
3883:
3882:
3866:
3863:
3862:
3846:
3843:
3842:
3807:
3804:
3803:
3772:
3769:
3768:
3738:
3718:
3716:
3693:
3690:
3689:
3648:
3645:
3644:
3628:
3609:
3608:
3587:
3583:
3562:
3558:
3537:
3533:
3512:
3508:
3491:
3474:
3457:
3452:
3428:
3424:
3418:
3417:
3392:
3388:
3370:
3366:
3365:
3361:
3340:
3336:
3315:
3311:
3307:
3303:
3285:
3281:
3264:
3247:
3230:
3225:
3219:
3218:
3198:
3194:
3176:
3172:
3171:
3167:
3150:
3129:
3125:
3108:
3091:
3086:
3080:
3079:
3053:
3049:
3048:
3044:
3027:
3006:
3002:
2997:
2991:
2990:
2964:
2960:
2959:
2955:
2950:
2926:
2922:
2915:
2913:
2910:
2909:
2879:
2864:
2849:
2834:
2832:
2829:
2828:
2800:
2796:
2779:
2762:
2741:
2737:
2732:
2729:
2728:
2702: and
2699:
2695:
2691:
2674:
2657:
2615:
2611:
2607:
2602:
2599:
2598:
2595:
2565:
2550:
2535:
2489:
2478:
2475:
2474:
2306:
2295:
2292:
2291:
2260:
2257:
2256:
2200: and
2197:
2186:
2183:
2182:
2151:
2148:
2147:
2125:
2119:
2089:
2074:
2059:
2057:
2054:
2053:
1970: and
1967:
1956:
1953:
1936:If two events,
1934:
1896:
1893:
1892:
1856:
1853:
1850:
1849:
1847:
1840:
1837:
1834:
1833:
1831:
1829:
1826:
1807:
1788:
1786:
1783:
1782:
1756:
1752:
1739:
1737:
1734:
1733:
1716:
1712:
1701:
1699:
1696:
1695:
1651:
1649:
1646:
1645:
1620:
1617:
1616:
1591:
1588:
1587:
1543:
1540:
1539:
1532:
1518:
1480:games of chance
1476:Punnett squares
1439:assessment and
1433:
1370:
1364:
1306:Hermann Laurent
1218:
1215:
1214:
1198:
1195:
1194:
1173:
1169:
1163:
1159:
1155:
1151:
1131:
1128:
1127:
1033:Ars Conjectandi
1028:Jakob Bernoulli
966:Richard Jeffrey
954:
948:
942:
894:from the Latin
885:
879:
869:
784:
778:
776:Interpretations
770:complex systems
685:
640:
639:
590:
582:
581:
577:Decision theory
525:
479:
327:Random variable
278:Bernoulli trial
155:
115:
114:
69:
54:
47:
28:
23:
22:
15:
12:
11:
5:
7195:
7185:
7184:
7179:
7164:
7163:
7140:
7139:
7137:
7136:
7131:
7126:
7121:
7116:
7111:
7106:
7101:
7096:
7091:
7089:Nanotechnology
7086:
7081:
7076:
7071:
7066:
7061:
7059:Machine vision
7056:
7051:
7046:
7045:
7044:
7039:
7034:
7029:
7024:
7016:
7015:
7014:
7009:
7001:
6996:
6991:
6986:
6985:
6984:
6979:
6971:
6966:
6961:
6956:
6951:
6946:
6941:
6936:
6931:
6926:
6921:
6916:
6911:
6906:
6901:
6896:
6891:
6886:
6881:
6875:
6872:
6871:
6862:Glossaries of
6859:
6858:
6851:
6844:
6836:
6827:
6826:
6824:
6823:
6818:
6808:
6803:
6790:
6787:
6786:
6783:
6782:
6780:
6779:
6774:
6769:
6764:
6759:
6753:
6751:
6747:
6746:
6744:
6743:
6738:
6733:
6727:
6725:
6716:
6712:
6711:
6709:
6708:
6703:
6698:
6693:
6688:
6683:
6678:
6673:
6668:
6663:
6658:
6653:
6648:
6643:
6642:
6641:
6631:
6626:
6621:
6616:
6611:
6610:
6609:
6604:
6594:
6589:
6584:
6579:
6573:
6571:
6567:
6566:
6563:
6562:
6560:
6559:
6554:
6549:
6544:
6538:
6536:
6532:
6531:
6529:
6528:
6523:
6518:
6513:
6512:
6511:
6506:
6496:
6490:
6488:
6481:
6480:
6475:
6470:
6465:
6460:
6455:
6450:
6444:
6442:
6438:
6437:
6435:
6434:
6429:
6423:
6420:
6419:
6412:
6411:
6404:
6397:
6389:
6383:
6382:
6377:
6354:
6324:
6311:
6306:
6297:
6292:
6287:
6269:
6264:
6249:
6240:
6239:
6233:
6232:
6221:
6220:
6218:
6217:External links
6215:
6214:
6213:
6198:
6184:
6170:Kallenberg, O.
6165:
6162:
6159:
6158:
6151:
6129:
6113:
6092:
6080:
6055:
6046:
6037:
6028:
6019:
5990:
5983:
5965:
5938:
5903:
5829:
5820:
5791:
5773:
5746:
5724:
5693:
5677:
5664:(4): 352–355.
5644:
5637:
5619:
5612:
5594:
5565:
5557:magazine 2012"
5553:Gorrochum, P.
5540:
5527:
5507:
5483:
5457:
5450:
5432:
5425:
5407:
5400:
5382:
5355:
5328:
5321:
5300:
5288:
5267:
5249:
5248:
5246:
5243:
5240:
5239:
5229:
5181:
5180:
5178:
5175:
5174:
5173:
5168:
5163:
5158:
5148:Main article:
5146:
5145:
5131:
5115:
5112:
4987:Main article:
4984:
4981:
4978:
4977:
4962:
4959:
4956:
4953:
4948:
4945:
4942:
4939:
4936:
4933:
4929:
4925:
4922:
4919:
4913:
4907:
4904:
4901:
4898:
4893:
4890:
4887:
4884:
4881:
4878:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4844:
4840:
4839:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4787:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4744:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4709:
4705:
4702:
4699:
4696:
4693:
4691:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4670:
4660:
4656:
4655:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4600:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4523:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4502:
4492:
4488:
4487:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4454:
4449:
4445:
4441:
4438:
4428:
4424:
4423:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4375:
4371:
4370:
4367:
4357:
4354:
4341:
4321:
4301:
4298:
4294:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4259:
4255:
4252:
4249:
4225:
4203:
4199:
4176:
4172:
4151:
4148:
4124:
4119:
4115:
4092:
4088:
4058:Main article:
4055:
4052:
4040:
4036:
4032:
4012:
4009:
4005:
4001:
3980:
3977:
3973:
3969:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3870:
3850:
3826:
3823:
3820:
3817:
3814:
3811:
3791:
3788:
3785:
3782:
3779:
3776:
3750:
3747:
3744:
3741:
3736:
3733:
3730:
3727:
3724:
3721:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3667:
3664:
3661:
3658:
3655:
3652:
3627:
3624:
3606:
3602:
3599:
3596:
3593:
3590:
3586:
3582:
3579:
3575:
3571:
3568:
3565:
3561:
3557:
3554:
3550:
3546:
3543:
3540:
3536:
3532:
3529:
3525:
3521:
3518:
3515:
3511:
3507:
3504:
3500:
3497:
3494:
3490:
3487:
3483:
3480:
3477:
3473:
3470:
3466:
3463:
3460:
3456:
3453:
3451:
3447:
3443:
3440:
3437:
3434:
3431:
3427:
3423:
3420:
3419:
3415:
3410:
3405:
3401:
3398:
3395:
3391:
3387:
3383:
3379:
3376:
3373:
3369:
3364:
3360:
3357:
3353:
3349:
3346:
3343:
3339:
3335:
3332:
3328:
3324:
3321:
3318:
3314:
3310:
3306:
3302:
3298:
3294:
3291:
3288:
3284:
3280:
3277:
3273:
3270:
3267:
3263:
3260:
3256:
3253:
3250:
3246:
3243:
3239:
3236:
3233:
3229:
3226:
3224:
3221:
3220:
3216:
3211:
3207:
3204:
3201:
3197:
3193:
3189:
3185:
3182:
3179:
3175:
3170:
3166:
3163:
3159:
3156:
3153:
3149:
3146:
3142:
3138:
3135:
3132:
3128:
3124:
3121:
3117:
3114:
3111:
3107:
3104:
3100:
3097:
3094:
3090:
3087:
3085:
3082:
3081:
3077:
3073:
3070:
3066:
3062:
3059:
3056:
3052:
3047:
3043:
3040:
3036:
3033:
3030:
3026:
3023:
3019:
3015:
3012:
3009:
3005:
3001:
2998:
2996:
2993:
2992:
2988:
2984:
2981:
2977:
2973:
2970:
2967:
2963:
2958:
2954:
2951:
2949:
2945:
2941:
2938:
2935:
2932:
2929:
2925:
2921:
2918:
2917:
2893:
2887:
2884:
2878:
2872:
2869:
2863:
2857:
2854:
2848:
2842:
2839:
2813:
2809:
2806:
2803:
2799:
2795:
2792:
2788:
2785:
2782:
2778:
2775:
2771:
2768:
2765:
2761:
2758:
2754:
2750:
2747:
2744:
2740:
2736:
2716:
2712:
2708:
2698:
2694:
2690:
2687:
2683:
2680:
2677:
2673:
2670:
2666:
2663:
2660:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2628:
2624:
2618: or
2614:
2610:
2606:
2594:
2591:
2579:
2573:
2570:
2564:
2558:
2555:
2549:
2543:
2540:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2492: or
2488:
2485:
2482:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2309: or
2305:
2302:
2299:
2279:
2276:
2273:
2270:
2267:
2264:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2196:
2193:
2190:
2170:
2167:
2164:
2161:
2158:
2155:
2121:Main article:
2118:
2115:
2103:
2097:
2094:
2088:
2082:
2079:
2073:
2067:
2064:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1966:
1963:
1960:
1933:
1930:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1871:If two events
1795:
1791:
1770:
1767:
1764:
1759:
1755:
1751:
1746:
1743:
1719:
1715:
1711:
1707:
1704:
1664:
1661:
1658:
1633:
1630:
1627:
1624:
1604:
1601:
1598:
1595:
1586:is written as
1547:
1517:
1514:
1432:
1429:
1366:Main article:
1363:
1360:
1352:Artemas Martin
1337:measure theory
1283:probable error
1255:Morgan Crofton
1222:
1202:
1176:
1172:
1166:
1162:
1158:
1154:
1150:
1147:
1144:
1141:
1138:
1135:
1078:Thomas Simpson
986:(16th century)
944:Main article:
941:
938:
868:
865:
864:
863:
833:
780:Main article:
777:
774:
726:areas of study
687:
686:
684:
683:
676:
669:
661:
658:
657:
646:
645:
642:
641:
638:
637:
632:
627:
622:
617:
612:
607:
602:
597:
591:
588:
587:
584:
583:
580:
579:
570:
565:
556:
551:
542:
537:
532:
526:
521:
520:
513:
512:
511:
510:
505:
497:
496:
490:
489:
481:
480:
478:
477:
470:
463:
455:
452:
451:
450:
449:
444:
436:
435:
434:
433:
428:
426:Bayes' theorem
423:
418:
413:
408:
400:
399:
398:
397:
392:
387:
382:
374:
373:
372:
371:
370:
369:
364:
359:
357:Observed value
354:
349:
344:
342:Expected value
339:
334:
324:
319:
318:
317:
312:
307:
302:
297:
292:
282:
281:
280:
270:
269:
268:
263:
258:
253:
248:
238:
233:
225:
224:
223:
222:
217:
212:
211:
210:
200:
199:
198:
185:
184:
176:
175:
169:
168:
157:
156:
154:
153:
146:
139:
131:
128:
127:
126:
125:
112:
107:
102:
97:
92:
87:
82:
74:
73:
65:
64:
26:
9:
6:
4:
3:
2:
7194:
7183:
7180:
7178:
7175:
7174:
7172:
7162:
7157:
7152:
7151:
7148:
7135:
7132:
7130:
7127:
7125:
7122:
7120:
7117:
7115:
7112:
7110:
7107:
7105:
7102:
7100:
7097:
7095:
7092:
7090:
7087:
7085:
7082:
7080:
7077:
7075:
7072:
7070:
7067:
7065:
7062:
7060:
7057:
7055:
7052:
7050:
7047:
7043:
7040:
7038:
7035:
7033:
7030:
7028:
7025:
7023:
7020:
7019:
7017:
7013:
7010:
7008:
7005:
7004:
7002:
7000:
6997:
6995:
6992:
6990:
6987:
6983:
6980:
6978:
6975:
6974:
6972:
6970:
6967:
6965:
6962:
6960:
6957:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6900:
6897:
6895:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6876:
6873:
6869:
6865:
6857:
6852:
6850:
6845:
6843:
6838:
6837:
6834:
6822:
6819:
6816:
6812:
6809:
6807:
6804:
6802:
6792:
6791:
6788:
6778:
6777:Logic symbols
6775:
6773:
6770:
6768:
6765:
6763:
6760:
6758:
6755:
6754:
6752:
6748:
6742:
6739:
6737:
6734:
6732:
6729:
6728:
6726:
6724:
6720:
6717:
6713:
6707:
6704:
6702:
6699:
6697:
6694:
6692:
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6672:
6669:
6667:
6664:
6662:
6659:
6657:
6656:Logical truth
6654:
6652:
6649:
6647:
6644:
6640:
6637:
6636:
6635:
6632:
6630:
6627:
6625:
6622:
6620:
6617:
6615:
6612:
6608:
6605:
6603:
6600:
6599:
6598:
6597:Contradiction
6595:
6593:
6590:
6588:
6585:
6583:
6580:
6578:
6575:
6574:
6572:
6568:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6542:Argumentation
6540:
6539:
6537:
6533:
6527:
6526:Philosophical
6524:
6522:
6521:Non-classical
6519:
6517:
6514:
6510:
6507:
6505:
6502:
6501:
6500:
6497:
6495:
6492:
6491:
6489:
6485:
6479:
6476:
6474:
6471:
6469:
6466:
6464:
6461:
6459:
6456:
6454:
6451:
6449:
6446:
6445:
6443:
6439:
6433:
6430:
6428:
6425:
6424:
6421:
6417:
6410:
6405:
6403:
6398:
6396:
6391:
6390:
6387:
6381:
6378:
6375:
6374:88-8091-176-7
6371:
6367:
6366:
6361:
6355:
6353:
6351:
6346:
6342:
6339:
6335:
6331:
6328:
6325:
6323:
6319:
6315:
6312:
6310:
6307:
6305:
6301:
6298:
6296:
6293:
6291:
6288:
6285:
6281:
6277:
6273:
6270:
6268:
6265:
6263:
6259:
6258:
6253:
6250:
6248:
6245:
6244:
6238:
6235:
6234:
6229:
6224:
6211:
6210:0-471-67969-0
6207:
6203:
6199:
6197:
6196:0-387-95313-2
6193:
6189:
6185:
6183:
6182:0-387-25115-4
6179:
6175:
6171:
6168:
6167:
6154:
6148:
6144:
6140:
6133:
6126:
6122:
6117:
6108:
6103:
6096:
6090:
6084:
6069:
6065:
6059:
6050:
6041:
6032:
6023:
6008:
6004:
6003:"Probability"
5997:
5995:
5986:
5984:9780136033134
5980:
5976:
5969:
5961:
5957:
5953:
5949:
5942:
5934:
5930:
5926:
5922:
5919:(1): 93–106.
5918:
5914:
5907:
5899:
5897:
5893:
5889:
5883:
5879:
5874:
5869:
5865:
5861:
5857:
5853:
5849:
5848:
5843:
5839:
5833:
5824:
5816:
5810:
5802:
5798:
5794:
5788:
5784:
5777:
5761:
5760:CWI Quarterly
5757:
5750:
5742:
5735:
5728:
5712:
5708:
5704:
5697:
5690:
5684:
5682:
5672:
5667:
5663:
5659:
5655:
5648:
5640:
5634:
5630:
5623:
5615:
5609:
5605:
5598:
5582:
5578:
5577:
5569:
5558:
5556:
5552:
5544:
5537:
5531:
5525:
5524:0-521-39459-7
5521:
5517:
5511:
5504:
5500:
5496:
5492:
5491:Freund, John.
5487:
5481:
5477:
5473:
5469:
5464:
5462:
5453:
5447:
5443:
5436:
5428:
5422:
5418:
5411:
5403:
5397:
5393:
5386:
5370:
5366:
5359:
5351:
5347:
5343:
5339:
5332:
5324:
5318:
5314:
5310:
5304:
5297:
5292:
5285:
5284:0-471-25708-7
5281:
5277:
5271:
5264:
5260:
5254:
5250:
5233:
5226:
5222:
5218:
5214:
5210:
5206:
5202:
5200:
5195:
5193:
5186:
5182:
5172:
5169:
5167:
5164:
5162:
5159:
5157:
5154:
5153:
5151:
5143:
5132:
5129:
5123:
5118:
5111:
5109:
5105:
5101:
5097:
5093:
5089:
5085:
5081:
5077:
5073:
5069:
5065:
5061:
5060:wave function
5057:
5053:
5049:
5045:
5036:
5032:
5028:
5023:
5018:
5014:
5010:
5006:
5002:
5001:deterministic
4996:
4990:
4957:
4951:
4943:
4937:
4931:
4923:
4917:
4911:
4902:
4896:
4888:
4885:
4882:
4876:
4870:
4864:
4861:
4858:
4852:
4845:
4841:
4811:
4805:
4799:
4793:
4790:
4788:
4780:
4777:
4774:
4768:
4758:
4752:
4746:
4738:
4732:
4729:
4723:
4717:
4711:
4703:
4697:
4694:
4692:
4684:
4681:
4678:
4672:
4661:
4657:
4627:
4621:
4618:
4612:
4606:
4603:
4601:
4593:
4590:
4587:
4581:
4571:
4568:
4565:
4559:
4556:
4550:
4544:
4541:
4535:
4529:
4526:
4524:
4516:
4513:
4510:
4504:
4493:
4489:
4470:
4464:
4461:
4458:
4455:
4447:
4443:
4436:
4429:
4425:
4407:
4404:
4401:
4395:
4389:
4383:
4376:
4372:
4364:
4353:
4339:
4319:
4296:
4288:
4282:
4276:
4270:
4267:
4261:
4253:
4247:
4239:
4223:
4201:
4197:
4174:
4170:
4149:
4146:
4138:
4122:
4117:
4113:
4090:
4086:
4077:
4073:
4072:
4067:
4061:
4051:
4038:
4034:
4030:
4010:
4007:
4003:
3999:
3978:
3975:
3971:
3967:
3958:
3956:
3952:
3936:
3933:
3927:
3921:
3915:
3909:
3906:
3900:
3897:
3894:
3888:
3868:
3848:
3840:
3821:
3818:
3815:
3809:
3789:
3786:
3780:
3774:
3765:
3745:
3739:
3731:
3728:
3725:
3719:
3713:
3707:
3704:
3701:
3695:
3687:
3685:
3681:
3662:
3659:
3656:
3650:
3642:
3638:
3634:
3633:
3623:
3604:
3600:
3597:
3594:
3591:
3588:
3584:
3580:
3577:
3573:
3569:
3566:
3563:
3559:
3555:
3552:
3548:
3544:
3541:
3538:
3534:
3530:
3527:
3523:
3519:
3516:
3513:
3509:
3505:
3502:
3498:
3495:
3492:
3488:
3485:
3481:
3478:
3475:
3471:
3468:
3464:
3461:
3458:
3454:
3449:
3445:
3441:
3438:
3435:
3432:
3429:
3425:
3421:
3413:
3408:
3403:
3399:
3396:
3393:
3389:
3385:
3381:
3377:
3374:
3371:
3367:
3362:
3358:
3355:
3351:
3347:
3344:
3341:
3337:
3333:
3330:
3326:
3322:
3319:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3289:
3286:
3282:
3278:
3275:
3271:
3268:
3265:
3261:
3258:
3254:
3251:
3248:
3244:
3241:
3237:
3234:
3231:
3227:
3222:
3214:
3209:
3205:
3202:
3199:
3195:
3191:
3187:
3183:
3180:
3177:
3173:
3168:
3164:
3161:
3157:
3154:
3151:
3147:
3144:
3140:
3136:
3133:
3130:
3126:
3122:
3119:
3115:
3112:
3109:
3105:
3102:
3098:
3095:
3092:
3088:
3083:
3075:
3071:
3068:
3064:
3060:
3057:
3054:
3050:
3045:
3041:
3038:
3034:
3031:
3028:
3024:
3021:
3017:
3013:
3010:
3007:
3003:
2999:
2994:
2986:
2982:
2979:
2975:
2971:
2968:
2965:
2961:
2956:
2952:
2947:
2943:
2939:
2936:
2933:
2930:
2927:
2923:
2919:
2906:
2891:
2885:
2882:
2876:
2870:
2867:
2861:
2855:
2852:
2846:
2840:
2837:
2825:
2811:
2807:
2804:
2801:
2797:
2793:
2790:
2786:
2783:
2780:
2776:
2773:
2769:
2766:
2763:
2759:
2756:
2752:
2748:
2745:
2742:
2738:
2734:
2714:
2710:
2706:
2696:
2692:
2688:
2685:
2681:
2678:
2675:
2671:
2668:
2664:
2661:
2658:
2654:
2651:
2645:
2642:
2639:
2633:
2630:
2626:
2622:
2612:
2608:
2604:
2590:
2577:
2571:
2568:
2562:
2556:
2553:
2547:
2541:
2538:
2532:
2526:
2520:
2517:
2511:
2505:
2502:
2496:
2486:
2480:
2472:
2467:
2451:
2445:
2442:
2436:
2430:
2427:
2424:
2421:
2415:
2409:
2406:
2400:
2394:
2391:
2385:
2382:
2379:
2373:
2370:
2364:
2358:
2355:
2349:
2343:
2340:
2334:
2331:
2328:
2322:
2319:
2313:
2303:
2297:
2274:
2271:
2268:
2262:
2254:
2250:
2234:
2231:
2225:
2222:
2219:
2213:
2210:
2204:
2194:
2188:
2165:
2162:
2159:
2153:
2145:
2141:
2136:
2134:
2130:
2124:
2114:
2101:
2095:
2092:
2086:
2080:
2077:
2071:
2065:
2062:
2046:
2042:
2040:
2028:
2022:
2016:
2010:
2004:
2001:
1995:
1992:
1989:
1983:
1980:
1974:
1964:
1958:
1949:
1947:
1943:
1939:
1929:
1916:
1910:
1907:
1904:
1898:
1891:, denoted as
1890:
1886:
1882:
1878:
1874:
1869:
1867:
1822:
1818:
1814:
1810:
1793:
1789:
1768:
1762:
1757:
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1749:
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1454:
1450:
1446:
1442:
1438:
1428:
1426:
1422:
1417:
1415:
1411:
1410:Cox's theorem
1407:
1403:
1399:
1395:
1391:
1387:
1382:
1379:
1375:
1369:
1359:
1357:
1353:
1349:
1344:
1342:
1338:
1334:
1330:
1329:Markov chains
1326:
1325:Andrey Markov
1321:
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1286:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1235:John Herschel
1220:
1200:
1191:
1174:
1170:
1164:
1160:
1156:
1152:
1148:
1145:
1139:
1133:
1125:
1123:
1122:Robert Adrain
1119:
1115:
1111:
1107:
1099:
1095:
1093:
1089:
1087:
1082:
1079:
1075:
1071:
1067:
1062:
1060:
1057:
1053:
1049:
1045:
1044:
1039:
1035:
1034:
1029:
1025:
1021:
1020:Blaise Pascal
1017:
1012:
1008:
1004:
996:
992:
985:
981:
977:
975:
971:
967:
964:According to
962:
959:
953:
947:
937:
935:
931:
927:
923:
919:
915:
911:
907:
903:
898:
893:
890:
884:
878:
874:
861:
857:
853:
849:
845:
841:
837:
836:Subjectivists
834:
831:
827:
826:of occurrence
825:
820:
816:
813:
812:
811:
809:
805:
801:
797:
793:
789:
783:
773:
771:
767:
763:
759:
755:
751:
747:
743:
739:
735:
731:
727:
723:
722:
717:
712:
709:
705:
701:
693:
682:
677:
675:
670:
668:
663:
662:
660:
659:
656:
652:
648:
647:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
592:
586:
585:
578:
574:
571:
569:
566:
564:
560:
557:
555:
552:
550:
546:
543:
541:
538:
536:
533:
531:
530:Number theory
528:
527:
524:
519:
518:
515:
514:
509:
506:
504:
501:
500:
499:
498:
495:
492:
491:
487:
486:
476:
471:
469:
464:
462:
457:
456:
454:
453:
448:
445:
443:
440:
439:
438:
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432:
429:
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424:
422:
419:
417:
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412:
409:
407:
404:
403:
402:
401:
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393:
391:
388:
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378:
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375:
368:
365:
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343:
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330:
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328:
325:
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301:
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296:
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288:
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286:
283:
279:
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264:
262:
259:
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254:
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249:
247:
244:
243:
242:
239:
237:
234:
232:
229:
228:
227:
226:
221:
218:
216:
215:Indeterminism
213:
209:
206:
205:
204:
201:
197:
194:
193:
192:
189:
188:
187:
186:
182:
178:
177:
174:
171:
170:
167:
163:
162:
152:
147:
145:
140:
138:
133:
132:
130:
129:
124:
119:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
88:
86:
83:
81:
78:
77:
76:
75:
67:
66:
63:
60:
59:
56:
52:
45:
41:
37:
33:
19:
6973:Engineering
6924:Cell biology
6894:Architecture
6696:Substitution
6675:
6516:Mathematical
6441:Major fields
6363:
6347:
6316:pdf file of
6275:
6256:
6252:Probability
6227:
6201:
6187:
6173:
6164:Bibliography
6138:
6132:
6120:
6116:
6095:
6083:
6071:. Retrieved
6067:
6058:
6049:
6040:
6031:
6022:
6012:10 September
6010:. Retrieved
6006:
5974:
5968:
5951:
5947:
5941:
5916:
5912:
5906:
5895:
5891:
5887:
5885:
5851:
5845:
5832:
5823:
5782:
5776:
5764:. Retrieved
5759:
5749:
5740:
5727:
5715:. Retrieved
5711:the original
5706:
5696:
5661:
5657:
5647:
5628:
5622:
5603:
5597:
5585:. Retrieved
5581:the original
5575:
5568:
5554:
5550:
5543:
5535:
5530:
5515:
5510:
5497:. Dickenson
5494:
5486:
5471:
5441:
5435:
5416:
5410:
5391:
5385:
5373:. Retrieved
5368:
5358:
5341:
5337:
5331:
5312:
5309:Hacking, Ian
5303:
5291:
5275:
5270:
5253:
5232:
5224:
5220:
5216:
5205:sample space
5198:
5191:
5185:
5046:
5031:in principle
5030:
4998:
4369:Probability
4237:
4162:The odds on
4137:conditioning
4074:relates the
4069:
4063:
3959:
3837:is formally
3766:
3688:
3683:
3679:
3640:
3636:
3630:
3629:
2907:
2826:
2596:
2468:
2252:
2143:
2137:
2132:
2128:
2126:
2051:
1950:
1941:
1937:
1935:
1888:
1884:
1876:
1872:
1870:
1820:
1816:
1812:
1808:
1691:
1687:
1686:of an event
1683:
1679:
1677:
1583:
1578:
1570:
1536:sample space
1533:
1499:
1484:
1473:
1461:
1434:
1431:Applications
1418:
1383:
1371:
1347:
1345:
1322:
1318:George Boole
1310:Karl Pearson
1287:
1278:
1269:(1872), and
1192:
1126:
1117:
1113:
1104:
1090:
1083:
1073:
1063:
1058:
1051:
1041:
1031:
1000:
973:
969:
963:
955:
921:
897:probabilitas
888:
886:
822:
815:Objectivists
803:
800:well-defined
785:
719:
713:
699:
698:
567:
447:Tree diagram
442:Venn diagram
406:Independence
352:Markov chain
236:Sample space
190:
90:Probabilists
61:
55:
40:graph theory
7177:Probability
7161:Mathematics
7094:Ornithology
7079:Meteorology
7064:Mathematics
7054:Ichthyology
6889:Archaeology
6884:Agriculture
6868:engineering
6811:WikiProject
6681:Proposition
6676:Probability
6629:Description
6570:Foundations
6257:In Our Time
6228:Probability
6107:1008.1287v1
5468:Hacking, I.
5344:: 129–145.
5166:Fuzzy logic
5156:Contingency
5100:statistical
5072:determinism
4071:Bayes' rule
1946:independent
1491:automobiles
1487:reliability
1372:Like other
1251:W.F. Donkin
1243:James Ivory
1237:'s (1850).
1070:Roger Cotes
1048:Ian Hacking
922:probability
889:probability
792:experiments
762:game theory
734:mathematics
706:concerning
704:mathematics
700:Probability
620:Linguistics
610:Computation
605:Geosciences
568:Probability
494:Mathematics
362:Random walk
203:Determinism
191:Probability
62:Probability
36:game theory
7171:Categories
7109:Psychiatry
7018:Geography
6989:Entomology
6741:Set theory
6639:Linguistic
6634:Entailment
6624:Definition
6592:Consequent
6587:Antecedent
6320:(1963) at
5954:: iv–vii.
5901:(11 pages)
5766:27 January
5717:27 January
5245:References
5225:impossible
5096:discovered
4993:See also:
4989:Randomness
4843:A given B
2727:Rewritten,
1684:complement
1528:See also:
1504:and other
1469:groupthink
1386:Kolmogorov
974:approvable
970:probabilis
914:legal case
883:Likelihood
871:See also:
766:philosophy
730:statistics
630:Philosophy
573:Statistics
563:Set theory
273:Experiment
220:Randomness
166:statistics
44:statistics
6964:Economics
6929:Chemistry
6904:Astronomy
6772:Fallacies
6767:Paradoxes
6757:Logicians
6691:Statement
6686:Reference
6651:Induction
6614:Deduction
6577:Abduction
6547:Metalogic
6494:Classical
6458:Inference
5933:144540412
5896:Mendelism
5888:Mendelism
5809:cite book
5801:949759319
5762:(1): 3–18
5691:, 18, 143
5201:certainly
5082:famously
5005:Newtonian
4886:∩
4862:∣
4778:∩
4682:∩
4591:∪
4569:∩
4557:−
4514:∪
4462:−
4448:∁
4396:∈
4268:∝
4189:to event
4105:to event
4078:of event
3951:σ-algebra
3898:∩
3839:undefined
3819:∣
3729:∩
3705:∣
3660:∣
3598:∩
3592:∩
3567:∩
3553:−
3542:∩
3528:−
3517:∩
3503:−
3439:∪
3433:∪
3397:∩
3386:∩
3375:∩
3356:−
3345:∩
3320:∩
3301:−
3290:∩
3276:−
3203:∩
3192:∪
3181:∩
3162:−
3134:∩
3120:−
3069:∩
3058:∪
3039:−
3011:∪
2980:∪
2969:∪
2937:∪
2931:∪
2862:−
2805:∩
2791:−
2746:∪
2686:−
2643:∪
2422:−
2383:∩
2371:−
2332:∪
2272:∪
2223:∩
2163:∩
2131:or event
2072:×
1993:∩
1908:∩
1790:∼
1766:¬
1758:∁
1745:¯
1565:power set
1560:power set
1546:Ω
1343:in 1931.
1323:In 1906,
1263:De Morgan
1157:−
1134:ϕ
1005:polymath
906:authority
887:The word
867:Etymology
794:that are
716:axiomatic
635:Education
625:Economics
600:Chemistry
266:Singleton
7134:Virology
7119:Robotics
7084:Mycology
7074:Medicine
6919:Calculus
6806:Category
6706:Validity
6607:Antinomy
6535:Theories
6499:Informal
6341:Archived
6330:Archived
5882:22964834
5858:: 3–13.
5847:Genetics
5493:(1973)
5375:22 April
5311:(1965).
5114:See also
5088:Max Born
5084:remarked
5017:roulette
4659:A and B
3682:, given
1815:) = 1 −
1706:′
1680:opposite
1495:warranty
1441:modeling
1374:theories
1300:(1853),
1273:(1875).
1267:Glaisher
1265:(1864),
1261:(1844),
1249:(1838),
1022:(1654).
972:) meant
958:Gambling
918:nobility
790:– i.e.,
746:gambling
728:such as
549:Analysis
545:Calculus
535:Geometry
347:Variance
110:Category
105:Journals
100:Notation
95:Glossary
18:Probable
7099:Physics
7049:Geology
6959:Ecology
6909:Biology
6864:science
6821:changes
6813: (
6671:Premise
6602:Paradox
6432:History
6427:Outline
6260:at the
6172:(2005)
5873:3430543
5470:(2006)
5211:on the
5104:reality
5052:physics
4491:A or B
1860:
1848:
1844:
1832:
1445:markets
1406:measure
1290:Laplace
1003:Italian
940:History
910:witness
902:probity
892:derives
742:finance
738:science
615:Biology
595:Physics
540:Algebra
503:History
261:Outcome
80:Outline
7147:Portal
6914:Botany
6723:topics
6509:Reason
6487:Logics
6478:Syntax
6372:
6338:Source
6322:UbuWeb
6225:about
6208:
6194:
6180:
6149:
6073:4 July
5981:
5931:
5880:
5870:
5799:
5789:
5635:
5610:
5587:23 May
5555:Chance
5522:
5505:(p. 1)
5501:
5478:
5448:
5423:
5398:
5319:
5282:
5261:
5199:almost
5192:almost
5094:, who
4427:not A
4366:Event
2253:either
1558:. The
1402:events
1376:, the
1362:Theory
1354:. See
1281:, the
1275:Peters
1193:where
875:, and
796:random
764:, and
708:events
208:System
196:Axioms
6750:other
6715:Lists
6701:Truth
6468:Proof
6416:Logic
6102:arXiv
5929:S2CID
5737:(PDF)
5560:(PDF)
5237:game.
5194:never
5177:Notes
4999:In a
3802:then
1811:(not
1781:, or
1644:, or
1581:event
1259:Ellis
1239:Gauss
912:in a
908:of a
559:Logic
523:Areas
508:Index
241:Event
42:, or
6866:and
6815:talk
6661:Name
6646:Form
6370:ISBN
6282:and
6206:ISBN
6192:ISBN
6178:ISBN
6147:ISBN
6075:2022
6014:2020
5979:ISBN
5878:PMID
5815:link
5797:OCLC
5787:ISBN
5768:2016
5719:2016
5633:ISBN
5608:ISBN
5589:2008
5520:ISBN
5499:ISBN
5476:ISBN
5446:ISBN
5421:ISBN
5396:ISBN
5377:2013
5317:ISBN
5280:ISBN
5259:ISBN
5213:real
5038:6.02
4076:odds
3861:and
2144:both
1944:are
1940:and
1887:and
1875:and
1830:1 −
1678:The
1500:The
1447:use
1437:risk
1398:sets
1316:and
1064:The
1054:and
1018:and
1011:odds
932:and
798:and
575:and
561:and
547:and
7027:N–Z
7022:A–M
7012:M–Z
7007:0–L
6982:M–Z
6977:A–L
6557:Set
6302:on
6284:PDF
6262:BBC
6254:on
5956:doi
5921:doi
5868:PMC
5860:doi
5852:192
5666:doi
5346:doi
5221:not
4064:In
3957:).
3767:If
2473:is
2471:die
2290:and
2181:and
1883:of
1682:or
1423:or
1396:),
1390:Cox
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1050:'s
1040:'s
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7173::
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6066:.
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4784:)
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4411:]
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4399:[
4393:)
4390:A
4387:(
4384:P
4340:B
4320:A
4300:)
4297:A
4293:|
4289:B
4286:(
4283:P
4280:)
4277:A
4274:(
4271:P
4265:)
4262:B
4258:|
4254:A
4251:(
4248:P
4224:A
4202:2
4198:A
4175:1
4171:A
4150:.
4147:B
4123:,
4118:2
4114:A
4091:1
4087:A
4035:/
4031:2
4011:,
4008:3
4004:/
4000:1
3979:;
3976:2
3972:/
3968:1
3934:=
3931:)
3928:B
3925:(
3922:P
3919:)
3916:A
3913:(
3910:P
3907:=
3904:)
3901:B
3895:A
3892:(
3889:P
3869:B
3849:A
3825:)
3822:B
3816:A
3813:(
3810:P
3790:0
3787:=
3784:)
3781:B
3778:(
3775:P
3749:)
3746:B
3743:(
3740:P
3735:)
3732:B
3726:A
3723:(
3720:P
3714:=
3711:)
3708:B
3702:A
3699:(
3696:P
3684:B
3680:A
3666:)
3663:B
3657:A
3654:(
3651:P
3641:B
3637:A
3605:)
3601:C
3595:B
3589:A
3585:(
3581:P
3578:+
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3570:C
3564:B
3560:(
3556:P
3549:)
3545:C
3539:A
3535:(
3531:P
3524:)
3520:B
3514:A
3510:(
3506:P
3499:)
3496:C
3493:(
3489:P
3486:+
3482:)
3479:B
3476:(
3472:P
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3462:A
3459:(
3455:P
3450:=
3446:)
3442:C
3436:B
3430:A
3426:(
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3409:)
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3394:B
3390:(
3382:)
3378:C
3372:A
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3352:)
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3342:B
3338:(
3334:P
3331:+
3327:)
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3317:A
3313:(
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3305:(
3297:)
3293:B
3287:A
3283:(
3279:P
3272:)
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3259:+
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3252:B
3249:(
3245:P
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3235:A
3232:(
3228:P
3223:=
3215:)
3210:)
3206:C
3200:B
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3188:)
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3178:A
3174:(
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3158:)
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3152:(
3148:P
3145:+
3141:)
3137:B
3131:A
3127:(
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3116:)
3113:B
3110:(
3106:P
3103:+
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3096:A
3093:(
3089:P
3084:=
3076:)
3072:C
3065:)
3061:B
3055:A
3051:(
3046:(
3042:P
3035:)
3032:C
3029:(
3025:P
3022:+
3018:)
3014:B
3008:A
3004:(
3000:P
2995:=
2987:)
2983:C
2976:)
2972:B
2966:A
2962:(
2957:(
2953:P
2948:=
2944:)
2940:C
2934:B
2928:A
2924:(
2920:P
2892:,
2877:=
2868:3
2847:+
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2808:B
2802:A
2798:(
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2787:)
2784:B
2781:(
2777:P
2774:+
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2767:A
2764:(
2760:P
2757:=
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2743:A
2739:(
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2715:.
2711:)
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2697:A
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2682:)
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2669:+
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2662:A
2659:(
2655:P
2652:=
2649:)
2646:B
2640:A
2637:(
2634:P
2631:=
2627:)
2623:B
2613:A
2609:(
2605:P
2578:.
2572:3
2569:1
2563:=
2557:6
2554:1
2548:+
2542:6
2539:1
2533:=
2530:)
2527:2
2524:(
2521:P
2518:+
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2512:1
2509:(
2506:P
2503:=
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2497:2
2487:1
2484:(
2481:P
2455:)
2452:B
2449:(
2446:P
2443:+
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2437:A
2434:(
2431:P
2428:=
2425:0
2419:)
2416:B
2413:(
2410:P
2407:+
2404:)
2401:A
2398:(
2395:P
2392:=
2389:)
2386:B
2380:A
2377:(
2374:P
2368:)
2365:B
2362:(
2359:P
2356:+
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2350:A
2347:(
2344:P
2341:=
2338:)
2335:B
2329:A
2326:(
2323:P
2320:=
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2314:B
2304:A
2301:(
2298:P
2278:)
2275:B
2269:A
2266:(
2263:P
2235:0
2232:=
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2226:B
2220:A
2217:(
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2211:=
2208:)
2205:B
2195:A
2192:(
2189:P
2169:)
2166:B
2160:A
2157:(
2154:P
2133:B
2129:A
2102:.
2096:4
2093:1
2087:=
2081:2
2078:1
2066:2
2063:1
2029:.
2026:)
2023:B
2020:(
2017:P
2014:)
2011:A
2008:(
2005:P
2002:=
1999:)
1996:B
1990:A
1987:(
1984:P
1981:=
1978:)
1975:B
1965:A
1962:(
1959:P
1942:B
1938:A
1917:.
1914:)
1911:B
1905:A
1902:(
1899:P
1889:B
1885:A
1877:B
1873:A
1862:.
1857:6
1854:/
1851:5
1841:6
1838:/
1835:1
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1821:A
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1817:P
1813:A
1809:P
1794:A
1769:A
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1754:A
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1742:A
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1714:A
1710:,
1703:A
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1657:(
1632:)
1629:A
1626:(
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1603:)
1600:A
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1584:A
1279:r
1221:c
1201:h
1175:2
1171:x
1165:2
1161:h
1153:e
1149:c
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1140:x
1137:(
1116:(
680:e
673:t
666:v
474:e
467:t
460:v
150:e
143:t
136:v
53:.
46:.
20:)
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