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1

,( , (),.. ,().

2 (1), ( ),.,(2 ) (2),,o (3 )(3)

: 4

.

(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

.

n

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 ,.

( ).)

: 36

. .,:

0,017

0 00056 0,00056

100 0,00035

0,000042

0,000035

0,000000004

6

: 37

BrianEveritt (1999). ,

. (1978). ( ) . (1995) , (1995).,,.

: 38