( ) ( C) 1 probability probability probable probable probable probable probably probably maybe perhaps possibly likely possibly<maybe perhaps<likely<probably probably ( ) 1 probably = almost surely * (2013 3 24 ) 1 Macmillan Dictionary for Children, Robert B. Costello (Ed.), Simon & Schuster, 2001.
2 probably 3 probability theory probability theory (gàil`ü) 2 1960 2.1, 1: 583 589 666 1853 2967 2 3 1973
4 1 5 5 100 100 40054 ZIP 6 ZIP tar gz π π π 2.2... 0 1 {0, 1}- ( 2) {0, 1}- 1 {0, 1}- {0, 1}- {0, 1}- {0, 1}- 4 5 bitmap 6
2: CD DVD 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 ( ) ( ) 1 0 1 x 2 + y 2 = 1 x 2 + y 2 = 1 3 1 0 {0, 1}- {0, 1}- {0, 1}- 10000 {0, 1}- 10000 5000 5000 { }} { { }} { 0, 0,..., 0, 1, 1,..., 1 (1) (1) 10000 {0, 1}- 10000 {0, 1}- {0, 1}- 10000 10000 {0, 1}- 2 10000 1/2 10000
10000 {0, 1}- 10000 {0, 1}- 9990 {0, 1}- 9990 {0, 1}- 2 9990 9989 {0, 1}- 2 9989 9990 {0, 1}- 2 9990 + 2 9989 + + 2 2 + 2 1 (2) (2) x 2x = 2 (2 9990 + 2 9989 + + 2 2 + 2 1 ) = 2 9991 + 2 9990 + + 2 3 + 2 2 = 2 9991 + x 2 1, x x = 2 9991 2 10000 {0, 1}- 2 10000 9990 {0, 1}- 2 9991 2 2 10000 < 29991 2 10000 = 1 2 9 = 1 512 10000 {0, 1}- 9990 {0, 1}- 511/512 ; 1 {0, 1}- 10000 2 10000 1 10000 511/512 9990 ; 2 {0, 1}- 1 1 2 {0, 1}-
4 4.1 {0, 1}- {0, 1}- 1 probability theory 1 1 3: ( ) {0,1} - 1 {0,1} - 1
4.2 1/2 1 100 1 0 1110110101 1011101101 0100000011 0110101001 0101000100 0101111101 1010000000 1010100011 0100011001 1101111101 100 1 100 51 50 100 k 100C k = 100!, k = 0, 1,..., 100 (100 k)! k! 50 ± 15 35 65 65 k=35 100C k = 1265381593893094343173016682006 100 2 100 = 1267650600228229401496703205376 100 35 65 65 100C k k=35 = 1265381593893094343173016682006 2 100 1267650600228229401496703205376 = 0.9982100696 99.8% 50 ± 15 10 ; 1000 500 ± 150 350 650 650 1000C k k=350 2 1000 = 0.999999999999999999999135703693 1 500 ± 50 450 550 550 1000C k k=450 2 1000 = 0.998608258405577913777674858896
99.9% 10000 5000 ± 150 4850 5150 5150 10000C k k=4850 99.9% 2 10000 = 0.998694631046660666521012366458 (3) (PC) (3) PC 7 PC PC / 100 50 ± 15 0.30 99.8% 1000 500 ± 50 0.10 99.9% 10000 5000 ± 150 0.03 99.9% / 10000 0 10000 3% 5000 ± 150 99.9% 1 ε > 0 n (1/2) ± ε 1 10000 {0, 1}- {0, 1}- 99.9% 1 5000 ± 150 10000 1 5000 ± 150 ε > 0 n n 1 (1/2) ± ε 1/2 2 ( 1713 ) p 1 p ε > 0 n p ± ε 1 7 MacBook Pro 2.7GHz Intel Core i7 + Mathematica 9
; n p ± ε 1 (1/4nε 2 ) P ( n ) p < ε 1 1 4nε 2. (4) 4.3 1/2 1/2 5 5.1 100 6 p p 100 10 6 6 S
100 6 S p 10 6 n S/10 6 p ± ε 1 P ( ) S 10 p < ε 6 1 1 4 10 6 ε 2, ε = 1/200 P ( S 10 p < 1 ) 6 200 99 100 (5) S 100 10 6 = 10 8 {0, 1}- 10 8 S/10 6 (5) S/10 6 99% 1/200 p 5.2 (5) 10 8 {0, 1}- ω ω 8 10 8 ω 10 8 {0, 1}- ω ω ω {0, 1}- ω {0, 1}- {0, 1}- {0, 1}- {0, 1}- {0, 1}- 8
5.3 g : {0, 1} l {0, 1} n, l < n {0, 1} l {0, 1} n l n {0, 1}- l ω {0, 1} l ω l {0, 1}- g(ω ) {0, 1} n g(ω ) ω g(ω ) n ω ω {0, 1} l ω = g(ω ) {0, 1} n S(ω) p 10 6 < 1 200 {0, 1}- ω ω g : {0, 1} 238 {0, 1} 108 ω {0, 1} 238 (5) ( S(g(ω )) P p 10 6 < 1 ) 200 99 100 ( - - (RWS ) 4) ω {0, 1} 238 238 ω g p {0, 1}- 9 S {0, 1} 108 9 H. Sugita Monte Carlo method, random number, and pseudorandom number, MSJ Memoirs vol.25 Chapter 2
4: RWS g : {0, 1} 238 {0, 1} 108 ( ) ω' ; S(ω) = 10 6 k=1 X(ω 100(k 1)+1,..., ω 100k ), ω = (ω 1,..., ω 10 8) {0, 1} 108, X (ξ 1,..., ξ 100 ) {0, 1} 100 { 1 (ξ1,..., ξ 100 1 6 ), X(ξ 1,..., ξ 100 ) = 0 ( ), RWS g : {0, 1} 238 {0, 1} 108 m = 100 N = 10 6 j = log 2 N = 19 238 = 2m + 2j 10 8 = Nm g : {0, 1} 2m+2j {0, 1} Nm ω = (ω 1,..., ω 2m+2j) {0, 1} 2m+2j x = m+j i=1 2 i ω i, α = m+j i=1 2 i ω m+j+i Z k = (d 1 (x + kα),..., d m (x + kα)) {0, 1} m, k = 1,..., N d i (t) t 0 2 i (0 1) Z k ω g(ω ) = (Z 1,..., Z N ) {0, 1} Nm
ω {0, 1} 238 1110110101 1011101101 0100000011 0110101001 0101000100 0101111101 1010000000 1010100011 0100011001 1101111101 1101010011 1111001001 1000001110 1110001000 0011010111 0010000010 0100010001 0101011011 1101011100 0100100111 0000000110 1010001100 1011100100 10111111 S(g(ω )) = 546177 p 0.546177 10 6... 6.1 6.2 vs 10 PC
11... 2... 12 ; probability mathematics µαθηµατικóς ( ) http://www.math.osaka-u.ac.jp/~sugita/mcm.html 11 8 2006 12 1991 6 7 21