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(384322..)
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2 (1), ( ),.,(2 ) (2),,o (3 )(3)
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.
(624621..) (6 6 ) (,).. (639559 )(639559..).
Thomas Aquinas (12251274..). .,.
Spinoza (16321677).
: 5
Girolamo Cardano (15011576), PierredeFermatBlaise Pascal(1654) Chevalier de Mr.
OChristiaan Huygens(1657).
: Cristjaan Huygens "De Ratiociniis in Aleae Ludo" (1657) :Cristjaan Huygens DeRatiociniis inAleae Ludo (1657),Jakob Bernoulli "Ars Conjectandi" (1713) AbrahamdeMoivre "DoctrineofChances" (1718) .
RogerCotes 1722ThomasSimpson 1756 ..
PierreSimonLaplace (1774). "TherieAnalytiquedesProbabilites(1795).
: 6
DanielBernoulli(1778)
. AdrienMarieLegendre(1805)
. RobertAdrain 1808,Legendre
. Gauss 1809 Gauss 1809
. Laplace(1810,1812),Gauss(1823),
JamesIvory(1825,1826),Hagen(1837),FriedrichBessel(1838), SimonDenisPoisson(1838), W.F.Donkin (1844,1856),andMorganCrofton(1870).
:PierreRmond deMontmort (1708),Ellis(1844),De ( ), ( ),Morgan(1864),Glaisher (1872),GiovanniSchiaparelli(1875) Peters(1856).
: 7
D.Hilbert 19001900.
von Mises (1919)'.
..Kolmogorov (1933). Kolmogorov .Kolmogorov ,. ,,,Bayes ..Kolmogorov [0,1]..Renyi 1955.
: 8
,
,1918ber reelle Funktionen(outermeasures) Caratheodory ExtensionTheorem KolmogorovKolmogorovExtensionTheorem.InfiniteDimensional Analysis Charalambos D Aliprantis Kim C BorderDimensional Analysis Charalambos D.Aliprantis KimC.Border(2005),10,1215,.
...,.
outermeasures
: 9
Pascal,Chevalier deMere (16071684), Mere (1607 1684),.
FermatPascal
Pascal Fermat (291654),Pascal:
:32pistoles ( 2 ) . 2).364pistoles.2 1,. pistoles ;pistoles ;
: 10
Pascal,.
48 (=32+16) 16 (=0+16)48( 32+16)16( 0+16)(3:1).
20,64,214816. 56 (=32+24) 8 (=0+8).56( 32 24)8( 0 8).
10,20 56 8 1 1 32 32568,113232.44(=28+16)20(=4+16).
: 11
FermatFermat3 2 ,2.4,, , :
A A A A B A A A B B B A B B B BA A A B A A B B A A B B A B B BA A B A A B A B A B A B B A B BA B A A A B B A B A A B B B A B
1 4 6 4 1
: 12
Roberval D'Alembert (
)4'4.
Pascal , 2 3 23,..
: 13
B
B
B
B
B
: 14
B
1 11n n m- + - 10 2m nk k + -=
1 11m n m- + 10
112m nk
n mk + -=
+ -
1 11 1:n mn m n m
k k- - + - + -
: 15
0 0k kk k= =
Pascal
1 011 1
1 2 11 3 3 1
1231 3 3 1
1 4 6 4 11 5 10 10 5 11 6 15 20 15 6 1
34561 6 15 20 15 6 1
. . . . . . . . . . . . . . .6
222 4222
m=3,n=4
: 16
: 17
: 18
(Galileo Galilei)9 10 10,.6.:
10 = 6 + 3 + 1= 6 + 2 + 2
9 = 6 + 2 +1= 5 + 3 +1
= 5 + 4 + 1= 5 + 3 + 2
= 5 + 2 + 2= 4 + 4 + 1 = 5 + 3 + 2
= 4 + 4 + 2= 4 + 4 + 1= 4 + 3 + 2
: 19
= 4 + 3 + 3= 3 + 3 + 3
Galileo Galilei (15641642)={1,2,3,4,5,6}
,
6 6 9 = 6 + 2 +1 10 = 6 + 3 + 13 6
6 3
= 5 + 3 +1= 5 + 2 + 2
= 6 + 2 + 2= 5 + 4 + 1 6
6 3 3
= 5 + 2 + 2= 4 + 4 + 1
= 5 + 4 + 1= 5 + 3 + 2
3 3
6 1
= 4 + 3 + 2= 3 + 3 + 3
= 4 + 4 + 2= 4 + 3 + 3
27 25
9 25 10 27 ( 216)
: 20
1307 1321 D t Ali li i " 13071321Dante Aliglieri "".1477Benvenuto d'Imola ,()(). ( ) 4:(1,1,2),(1,2,1),(2,1,1), C d Cardano ,1526(25)15761663,
.216. 4,2710259.
: 21
: 22
DeMr (1654)17 6(1).(6,6)24 (2)., 4 () 6 ( 1),4()6(1)24()36(2).O Ch li d M (1607 1684) OChevalier de Mr (16071684),1().2,.Pascal,Fermat.
: 23
6
0 518 0.518,1...
(6,6)
0 491 0.491,2.(!)DeMr,.
22425.
0 5050.505.
: 24
,,,
(rinciple of insufficient reason).
, , (equally likely),.
'.
()(), , .
.
: 25
: 26
: 27
"De Ratio ciniis in Ludo Aleae"
C.Hyugens (16291695),1657 Van
Huygens
VanSchooten. Christiaan Huygens
Huygens 1655,PascalFermat.
601700,.
Huygens yg .. Huygens " " Huygens .
.Fermat Pascal.C.Huygens Newton .
: 28
.
: 29
.
Von Mises . .','
..Hodges Lehman : 20250
1 2 0 276 0 372 p = 0 333120.2760.372,p =0.333. 2050.000
0.49710.5054,.0.0083, p 0 5p =0.5.
2019371956,230.000,
: 31
0.51260.5162.0.0036.
. .
2 56 56.
, . ,.1 ,1 .. .
: 32
n=10000000b=cumsum(2*(runif(n)>0.5)1)#plot(b,type="l")#abline(h 0)
n=100000b=cumsum(2*(runif(n)>0.5)1)#plot(b,type="l")#abline(h 0) #abline(h=0)
z[=100*sum(b>0)/n#z=99.89#abline(h=0)z=100*sum(b>0)/n#z=21.34
: 33
1000100000z0.5)-1)#plot(b,type="l")#abline(h=0)z[i]=100*sum(b>0)/n}hist(z,nclass=30)
: 34
: 35
649.
::1 6! 43! 1 0.000000071549 49! 139838166
0,0000000715. Douglas
6
0,0000000715.DouglasHofstadter ,.
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: 36
. .,:
0,017
0 00056 0,00056
100 0,00035
0,000042
0,000035
0,000000004
6
: 37
BrianEveritt (1999). ,
. (1978). ( ) . (1995) , (1995).,,.
: 38