Structure of the theories of probability

de Oliveira, Mário J.

doi:10.1590/1806-9126-RBEF-2022-0087

Mário J. de Oliveira ^** Correspondence email address: oliveira@if.usp.br

Universidade de São Paulo, Instituto de Física, 05508-090, São Paulo, SP, Brasil.

http://orcid.org/0000-0002-4139-1445

* Correspondence email address: oliveira@if.usp.br

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Figure 1
Two tables from a printed version of 1534 of the poem De Vetula [¹⁷[17] P. Ovidii Nasonis, De Vetula (1534).], written around 1250. The upper table shows the possible configurations of three dice and the possible outcome in the last column. The bottom table shows the possible outcomes in the first and second column, the number of configurations (punctaturae) and the number of ways of falling (cadentiae).

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Figure 2
A table from the book of Cardano on games of chance published in 1663 [²⁶[26] H. Cardani, Opera Omnia (Lugduni, 1663), p. 262.]. The upper part shows the number of cases in the throw of two dice. There is a misprint: at the right of the number 7 should be the number 6 and not 8. The lower part shows the number of cases in the throw of three dice.

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Figure 3
The arithmetic triangle from a figure of the treatise of Pascal on this subject [³³[33] B. Pascal, Traité du Triangle Arithmetic (Desprez, Paris, 1665).]. The number in a square is the sum of the number inside the upper and left squares in accordance with the property ().

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Figure 4
Table from the Huygens treatise [³⁷[37] F. Schooten, Exercitationum Mathematicarum (Elsevirii, Lugduni Batauorum, 1657).] showing the results for the problem of the division of a stake between three players. The upper part of each rectangle shows the numbers of rounds needed for winning the game. The lower part shows the respective share corresponding to each player.

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Figure 5
Table of combinations or figurative numbers from the Bernoullli treatise [⁴⁰[40] J. Bernoulli, Ars Conjectandi (Thurnisii fratres, Basileae, 1713).].

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Figure 6
Three ways of randomly drawing a chord on a circle. (a) Two points on the circumference are chosen at random. (b) One radius is chose and random and one point of the radius is randomly chosen to place the middle point of the chord. (c) A point inside the circle is chosen at random to be the middle point of the chord.

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aaaaaaabaabaaabbabaaabababbaabbbbaaabaabbabababbbbaabbabbbbabbbb

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aaaaabaacabaabbabcacaacbaccbaababbacbbabbbbbcbcabcbbcccaacabcaccbacbbcbcccaccbccc

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AAAAABABAABBBAABABBBABBB

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AAAAAAABAABAAABBABAAABABABBAABBBBAAABAABBABABABBBBAABBABBBBABBBB

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Table 1
Frequency of vowels and consonants in the Eugene Onegin according to Markov [⁸²[82] A.A. Markov, “Primer statisticheskogo issledovaniya nad tekstom ‘Evgeniya Onegina’, illyustriruyuschij svyaz’ ispytanij v cep’ ”, Izvestiya Akademii Nauk 7, 153 (1913)., ⁸³[83] A.A. Markov, “An example of statistical investigation of the text Eugene Onegin concerning the connection of samples in chains”, Science in Context 19, 591 (2006).]. The first table shows the frequencies of vowel (A) and the frequency of consonant (B). The second and third tables shows the frequencies of a vowel (A) or a consonant (B) following the group of letter shown in the firs row. We are indicating by A any vowel and by B any consonant.

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Table 2
Correspondence between sets and events. We are using the following notation:

\cap

and

\cup

for the union and intersection of sets instead of + and juxtaposition used by Kolmogorov. The empty set is denoted by V, and the complementary set of A by

\bar{A}

.

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Table 3
The author together with the abbreviated name of the main work where it is found, the year of its public presentation or publication, and references. Roman numerals indicate the century when the work appeared. Numbers between square brackets indicate the approximate date when the work was written.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

(92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

(100)

(101)

(102)

(103)

(104)

(105)

(106)

(107)

(108)

(109)

(110)

(111)

(112)

(113)

(114)

(115)

(116)

(117)

(118)

(119)

(120)

(121)

(122)

(123)

(124)

(125)

(126)

(127)

(128)

(129)

(130)

(131)

(132)

(133)

(134)

(135)

(136)

(137)

(138)

(139)

(140)

(141)

(142)

(143)

(144)

(145)

(146)

(147)

(148)

(149)

(150)

(151)

(152)

(153)

(154)

(155)

(156)

(157)

(158)

(159)

(160)

(161)

(162)

(163)

(164)

(165)

(166)

(167)

(168)

(169)

(170)

(171)

(172)

(173)

(174)

(175)

(176)

(177)

(178)

(179)

Events	Sets
collection of all elementary events	U
an elementary event	e, $e \in U$
an event	A, $A \subset U$
simultaneous occurrence of events	$A \cap B = X$
impossible event	V
incompatible events	$A \cap B = V$
occurrence of at least one event	$A \cup B = Y$
non-occurrence of event A	$\bar{A}$
event A follows inevitably from B	$B \subset A$

Author	Work	Year	Ref.
	De Vetula	xiii	[¹⁷[17] P. Ovidii Nasonis, De Vetula (1534).]
	commentary on Divine Comedy	xiv	[²⁰[20] A. Torri, L’Ottimo Commento della Divina Commedia (Niccolò Capurro, Pisa, 1827), 3 volumi.]
Cardano	Book on Games of Chance	[1550]	[²⁶[26] H. Cardani, Opera Omnia (Lugduni, 1663), p. 262.]
Galileo	Findings on the game of dice	[1620]	[²⁷[27] G. Galilei, in: Opere di Galileo Galilei (Tartini e Franchi, Firenze, 1718), v. 3, p. 119.]
Pascal and
Fermat	correspondence	1654	[³⁰[30] P. Tannery and C. Henry, Oeuvres de Fermat (Gauthier-Villars, Paris, 1894), v. 8, p. 288.]
Huygens	Calculation in Games of Chance	1657	[³⁷[37] F. Schooten, Exercitationum Mathematicarum (Elsevirii, Lugduni Batauorum, 1657).]
J. Bernoulli	Art of Conjecturing	1713	[⁴⁰[40] J. Bernoulli, Ars Conjectandi (Thurnisii fratres, Basileae, 1713).]
Montmort	Essay on the Games of Chance	1708	[⁴²[42] P.R. Montmort, Essay d’Analyze sur les Jeux de Hazards (Jacque Quillau, Paris, 1708).]
De Moivre	Doctrine of Chances	1718	[⁴⁵[45] A. Moivre, The Doctrine of Chances (Pearson, London, 1718).]
D. Bernoulli	conjectural problem	1770	[⁵²[52] D. Bernoulli, “Disquisitiones analyticae de novo problemate conjecturali”, Novi Commentarii Academiae Scientarum Imperialis Petropolitanae 14, 3 (1770).]
Lagrange	calculation of probability	1776	[⁵⁴[54] J.L. Lagrange, “Memoire sur l’utilite de la methode de prendre le milieu entre les resultats de plusiers observations”, Miscellanea Taurinensia 5, 167 (1776).]
Laplace	Theory of Probability	1812	[¹³[13] P.S. Laplace, Théorie Analytique des Probabilités (Courcier, Paris, 1812).]
Poisson	Research on Probability	1837	[⁴[4] S.D. Poisson, Recherches sur la Probablité de Jugements em Matière Criminelle et en Matière Civile (Bachelier, Paris, 1837).]
Bertrand	Calculus of Probability	1889	[⁷⁰[70] J. Bertrand, Calcul des Probabilités (Gauthier-Villars, Paris, 1889).]
Borel	Theory of Probability	1909	[⁷¹[71] E. Borel, Éléments de la Théorie des Probabilités (Hermann, Paris, 1909).]
Markov	dependent variables	1906	[⁷⁷[77] A.A. Markov, “Rasprostranenie zakona bol’shih chisel na velichiny zavisyaschie drug ot druga”, Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete 15, 135 (1906).]
Kolmogorov	Theory of Probability	1933	[⁸⁶[86] A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, Berlin, 1933).]

y = \frac{n}{3} + \frac{n}{\sqrt{3}} \sin \frac{r \sqrt{3}}{2 n} e^{- 3 r / 2 n} - \frac{n}{3} \cos \frac{r \sqrt{3}}{2 n} e^{- 3 r / 2 n} .

A = \sum_{j} \frac{n!}{(n + m - 2 j)! j! (j - m)!} a^{n + m - 2 j} b^{2 j - m},

A = a^{n} + n (n - 1) a^{n - 2} b^{2} + \frac{n (n - 1) (n - 2) (n - 3)}{2 \cdot 2} a^{n - 4} b^{4}

+ \frac{n (n - 1) (n - 2) (n - 3) (n - 4 (n - 5)}{2 \cdot 3 \cdot 2 \cdot 3} a^{n - 6} b^{6} + \dots .

\frac{1}{2} \int e^{- a^{2} x^{2} + i r x} d x + \frac{1}{2} \int e^{- a x^{2} - i r x} d x .

\int_{0}^{π / 2} \int_{0}^{ℓ \cos θ} \frac{2 d x d θ}{a π} = \frac{2 ℓ}{a π} \int_{0}^{π / 2} \cos θ d θ = \frac{2 ℓ}{a π} .

[1] * Correspondence email address: oliveira@if.usp.br

A	B	↱
0.128	0.663	A
0.872	0.337	B

AA	AB	BA	BB	↱
0.104	0.546	0.131	0.868	A
0.896	0.454	0.869	0.132	B


0.432	A
0.568	B