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than others. For example, if we consider the order of results there are six ways to obtain (621): (1,2,6), (1,6,2), (2,1,6), (2,6,1), (6,1,2), (6,2,1), but there is only one way to obtain (333), where the first, second and third dice all roll 3. There are a total of 27 permutations that sum to 10 but only 25 that sum to 9. From this, Cardano found that the probability of throwing a 9 is less than that of throwing a 10. He also demonstrated the efficacy of defining
1931:
1537:
1525:
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30:
413:, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).
369:
also thought about the sum of three dice. At face value there are the same number of combinations that sum to 9 as those that sum to 10. For a 9:(621) (531) (522) (441) (432) (333) and for 10: (631) (622) (541) (532) (442) (433). However, there are more ways of obtaining some of these combinations
363:(1265–1321) mentions this game. A commenter of Dante puts further thought into this game: the thought was that with three dice, the lowest number you can get is three, an ace for every die. Achieving a four can be done with three dice by having a two on one die and aces on the other two dice.
381:
wrote about die-throwing sometime between 1613 and 1623. Unknowingly considering what is essentially the same problem as
Cardano's, Galileo had said that certain numbers have the ability to be thrown because there are more ways to create that number.
264:
with the same sense), originally meaning "having the appearance of being strong or able" "having the similar appearance or qualities", with a meaning of "probably" recorded mid-15 c. The derived noun
528:, the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets, leading to the use of sophisticated probability models in
374:
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).
1244:
409:(1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental
268:
had a meaning of "similarity, resemblance" but took on a meaning of "probability" from the mid 15th century. The meaning "something likely to be true" is from 1570s.
959:
The Theory That Would Not Die: How Bayes' Rule
Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy
517:
that would be observed if the hypothesis is true. If observations approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected.
551:
was dominant, holding that probability means long-run relative frequency in a large number of trials. At the end of the century there was some revival of the
355:
invented dice during the Trojan wars, although their true origin is uncertain. The first dice game mentioned in literature of the
Christian era was called
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Probability deals with random experiments with a known distribution, Statistics deals with inference from the data about the unknown distribution.
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Games, gods and gambling: the origins and history of probability and statistical ideas from the earliest times to the
Newtonian era
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Electronic Journ@l for
History of Probability and Statistics/Journ@l Electronique d'Histoire des Probabilitéet de la Statistique
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view, according to which the fundamental notion of probability is how well a proposition is supported by the evidence for it.
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The mathematical treatment of probabilities, especially when there are infinitely many possible outcomes, was facilitated by
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provides evidence to show that the astragali were tossed into a circle drawn on the floor, much like playing marbles. In
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which explained properties of gases such as temperature in terms of the random motions of large numbers of particles.
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premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums.
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359:. Played with two or three dice, it was probably brought to Europe by the knights returning from the Crusades.
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Towards the end of the nineteenth century, a major success of explanation in terms of probabilities was the
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in which he consolidated and laid down many fundamental results in probability and statistics such as the
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234:(see probable). The mathematical sense of the term is from 1718. In the 18th century, the term
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Creating Modern
Probability: Its Mathematics, Physics and Philosophy in Historical Perspective
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A History of the
Mathematical Theory of Probability from the Time of Pascal to that of Laplace
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to correct error-prone observations, especially in astronomy, based on the assumption of a
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617:. For Cardano, see p. 54; for Fermat and Pascal, see pp. 59–61; for Huygens, see pp. 92–94
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The Lady
Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century
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in the 1560s (not published until 100 years later), and then in the correspondence
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Some laws and problems in classical probability and how
Cardano anticipated them
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Figures from the
History of Probability and Statistics (Univ. of Southampton)
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The mathematical methods of probability arose in the investigations first of
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Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
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The power of probabilistic methods in dealing with uncertainty was shown by
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A History of Probability and Statistics and Their Applications before 1750
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The Game of Probability. Literature and Calculation from Pascal and Kleist
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and their cognates in other modern languages derive from medieval learned
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Probability and statistics became closely connected through the work on
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The History of Statistics: The Measurement of Uncertainty before 1900
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given the evidence for them, and on the other hand the behavior of
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The field of the history of probability itself was established by
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The Science of Conjecture: Evidence and Probability Before Pascal
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140:. The study of the former is historically older in, for example,
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The theory of stochastic processes broadened into such areas as
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In ancient times there were games played using astragali, or
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The twentieth century also saw long-running disputes on the
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of errors to determine the most likely true value. In 1812,
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Earliest Uses of Symbols in Probability and Statistics
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A History of Mathematical Statistics from 1750 to 1930
870:(2nd ed.). New York: Cambridge University Press.
328:(1657) gave a comprehensive treatment of the subject.
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developed a grading of degrees of proof, credibility,
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to deal with the uncertainties of evidence in court.
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has a dual aspect: on the one hand the likelihood of
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252:is of Germanic origin, most likely from Old Norse
848:. Baltimore, MD: Johns Hopkins University Press.
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777:Against the Gods: The Remarkable Story of Risk
532:, including such successes as the widely used
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1069:Earliest Uses of Various Mathematical Symbols
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1023:. Belknap Press/Harvard University Press.
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824:. Princeton: Princeton University Press.
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301:times, betting was discussed in terms of
148:treatment of dice began with the work of
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666:Description of Greece. 1: Books I and II
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448:Théorie analytique des probabilités
228:) "credibility, probability," from
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305:such as "ten to one" and maritime
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802:. Stanford University Press.
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509:of drugs, as well as in
132:such as the throwing of
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978:von Plato, Jan (1994).
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399:(posthumous, 1713) and
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892:. Hoboken, NJ: Wiley.
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282:Ancient and medieval
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581:, pp. 113, 126.
538:valuation of options
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130:stochastic processes
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1470:Veterinary medicine
1164:Classical Antiquity
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914:. New York: Wiley.
780:. New York: Wiley.
772:Bernstein, Peter L.
721:, pp. 296–300.
691:, pp. 293–294.
643:. London: Griffin.
629:, pp. 296–316.
605:, pp. 278–288.
560:Kolmogorov's axioms
439:normal distribution
240:Doctrine of Chances
207:is from Old French
142:the law of evidence
45:Catalog of articles
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664:Pausanias (2007).
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1341:Combinatorics
1339:
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1309:
1308:Earth science
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1159:Ancient world
1157:
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1129:Pseudoscience
1127:
1125:
1122:
1120:
1117:
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1080:
1079:
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1043:
1042:
1032:
1030:0-674-40341-X
1026:
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1018:
1014:
1010:
1008:0-7167-4106-7
1004:
1000:
995:
991:
985:
981:
976:
972:
970:9780300169690
966:
961:
960:
953:
949:
947:0-387-95329-9
943:
939:
935:
931:
927:
923:
921:0-471-17912-4
917:
913:
909:
905:
901:
899:0-471-47129-1
895:
891:
887:
883:
879:
873:
869:
865:
861:
857:
855:0-8018-6569-7
851:
847:
846:
841:
837:
833:
831:0-691-08497-1
827:
823:
819:
815:
811:
809:9780804768641
805:
801:
797:
793:
789:
787:0-471-12104-5
783:
779:
778:
773:
769:
768:
757:, Chapter 18.
756:
751:
744:
739:
732:
727:
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709:
708:magazine 2012
707:
703:
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623:
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546:
541:
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534:Black–Scholes
531:
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508:
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500:
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377:In addition,
375:
373:
368:
364:
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329:
327:
323:
322:Blaise Pascal
319:
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127:
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112:
107:
105:
100:
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68:
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53:
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48:
46:
43:
41:
38:
37:
36:
35:
27:
26:
23:
20:
19:
16:
1807:Hindu-Arabic
1756:
1703:Group theory
1691:Trigonometry
1662:Topos theory
1480:Neuroscience
1384:Anthropology
1366:Trigonometry
1355:
1020:
998:
979:
958:
937:
930:Heyde, C. C.
911:
908:Hald, Anders
889:
886:Hald, Anders
867:
864:Hacking, Ian
844:
821:
799:
776:
750:
738:
726:
714:
705:
701:
696:
684:
665:
659:
640:
634:
622:
610:
598:
586:
574:
557:
542:
519:
503:Jerzy Neyman
499:R. A. Fisher
492:
482:
476:
461:
420:
404:
394:
389:
376:
365:
330:
311:
296:
288:presumptions
281:
265:
249:
239:
235:
225:probabilitas
222:(nominative
204:
200:
196:
179:
175:
174:
166:
146:mathematical
144:, while the
120:
50:Probabilists
15:
1956:Probability
1823:Mesopotamia
1797:Prehistoric
1757:Probability
1614:Algorithms
1447:Engineering
1389:Archaeology
1356:Probability
1323:Mathematics
1189:Romanticism
1174:Renaissance
549:frequentism
445:issued his
299:Renaissance
210:probabilite
205:probability
203:. The form
180:probability
122:Probability
22:Probability
1950:Categories
1747:Statistics
1679:Logarithms
1625:Arithmetic
1424:Technology
1409:Psychology
1361:Statistics
1208:By culture
1112:Background
934:Seneta, E.
566:References
333:talus bone
292:half-proof
276:See also:
266:likelihood
231:probabilis
188:probabilis
126:hypotheses
1767:Manifolds
1763:Topology
1674:Functions
1518:Timelines
1495:Pathology
1490:Nutrition
1414:Sociology
1394:Economics
1303:Chemistry
1293:Astronomy
1230:Byzantine
1225:Brazilian
1220:Argentine
511:economics
433:used the
353:Palamedes
349:Pausanias
307:insurance
197:plausible
171:Etymology
1935:Category
1910:timeline
1898:timeline
1772:timeline
1752:timeline
1740:timeline
1725:timeline
1713:timeline
1696:timeline
1686:Geometry
1657:timeline
1642:timeline
1637:Calculus
1630:timeline
1618:timeline
1608:timeline
1596:By topic
1588:timeline
1541:Category
1500:Pharmacy
1457:Medicine
1346:Geometry
1336:Calculus
1255:Japanese
1019:(1990).
910:(1998).
888:(2003).
866:(2006).
842:(2001).
820:(1988).
798:(2012).
774:(1996).
562:(1933).
553:Bayesian
485:(1865).
261:geliclic
245:cadentia
176:Probable
70:Category
65:Journals
60:Notation
55:Glossary
1802:Ancient
1603:Algebra
1475:Anatomy
1399:History
1331:Algebra
1313:Physics
1298:Biology
1275:Spanish
1270:Russian
1265:Mexican
1240:Chinese
1215:African
765:Sources
443:Laplace
379:Galileo
367:Cardano
335:. The
272:Origins
255:likligr
150:Cardano
40:Outline
1529:Portal
1260:Korean
1245:Indian
1235:French
1152:By era
1027:
1005:
986:
967:
944:
918:
896:
874:
852:
828:
806:
784:
706:Chance
672:
647:
357:Hazard
250:likely
236:chance
193:Cicero
158:Fermat
154:Pascal
1888:Other
1843:India
1838:China
1720:Logic
1351:Logic
427:Ceres
423:Gauss
341:Egypt
215:Latin
184:Latin
138:coins
1025:ISBN
1003:ISBN
984:ISBN
965:ISBN
942:ISBN
916:ISBN
894:ISBN
872:ISBN
850:ISBN
826:ISBN
804:ISBN
782:ISBN
670:ISBN
645:ISBN
524:and
501:and
470:and
372:odds
320:and
303:odds
290:and
178:and
160:and
134:dice
1067:on
497:of
466:of
403:'s
393:'s
297:In
199:or
136:or
1952::
932:;
540:.
351:,
156:,
152:,
1590:)
1586:(
1576:e
1569:t
1562:v
1097:e
1090:t
1083:v
1033:.
1011:.
992:.
973:.
950:.
924:.
902:.
880:.
858:.
834:.
812:.
790:.
745:.
678:.
653:.
593:.
110:e
103:t
96:v
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