2 probably 3 probability theory probability theory (gàil`ü) , 1:

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1 ( ) ( C) 1 probability probability probable probable probable probable probably probably maybe perhaps possibly likely possibly<maybe perhaps<likely<probably probably ( ) 1 probably = almost surely * ( ) 1 Macmillan Dictionary for Children, Robert B. Costello (Ed.), Simon & Schuster, 2001.

2 2 probably 3 probability theory probability theory (gàil`ü) , 1:

3 ZIP 6 ZIP tar gz π π π {0, 1}- ( 2) {0, 1}- 1 {0, 1}- {0, 1}- {0, 1}- {0, 1}- 4 5 bitmap 6

4 2: CD DVD ( ) ( ) x 2 + y 2 = 1 x 2 + y 2 = {0, 1}- {0, 1}- {0, 1} {0, 1} { }} { { }} { 0, 0,..., 0, 1, 1,..., 1 (1) (1) {0, 1} {0, 1}- {0, 1} {0, 1} /

5 10000 {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} (2) (2) x 2x = 2 ( ) = = x 2 1, x x = {0, 1} {0, 1} < = = {0, 1} {0, 1}- 511/512 ; 1 {0, 1} / ; 2 {0, 1} {0, 1}-

6 4 4.1 {0, 1}- {0, 1}- 1 probability theory 1 1 3: ( ) {0,1} - 1 {0,1} - 1

7 4.2 1/ k 100C k = 100!, k = 0, 1,..., 100 (100 k)! k! 50 ± k=35 100C k = = C k k=35 = = % 50 ± ; ± C k k= = ± C k k= =

8 99.9% ± C k k= % = (3) (PC) (3) PC 7 PC PC / ± % ± % ± % / % 5000 ± % 1 ε > 0 n (1/2) ± ε {0, 1}- {0, 1}- 99.9% ± ± 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

9 ; n p ± ε 1 (1/4nε 2 ) P ( n ) p < ε 1 1 4nε 2. (4) 4.3 1/2 1/ p p S

10 100 6 S p 10 6 n S/10 6 p ± ε 1 P ( ) S 10 p < ε ε 2, ε = 1/200 P ( S 10 p < 1 ) (5) S = 10 8 {0, 1} S/10 6 (5) S/ % 1/200 p 5.2 (5) 10 8 {0, 1}- ω ω ω 10 8 {0, 1}- ω ω ω {0, 1}- ω {0, 1}- {0, 1}- {0, 1}- {0, 1}- {0, 1}- 8

11 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 < {0, 1}- ω ω g : {0, 1} 238 {0, 1} 108 ω {0, 1} 238 (5) ( S(g(ω )) P p 10 6 < 1 ) ( - - (RWS ) 4) ω {0, 1} ω g p {0, 1}- 9 S {0, 1} H. Sugita Monte Carlo method, random number, and pseudorandom number, MSJ Memoirs vol.25 Chapter 2

12 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 ), X(ξ 1,..., ξ 100 ) = 0 ( ), RWS g : {0, 1} 238 {0, 1} 108 m = 100 N = 10 6 j = log 2 N = = 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

13 ω {0, 1} S(g(ω )) = p vs 10 PC

14 ; probability mathematics µαθηµατικóς ( )

~ ~.86 ~.02 ~.08 ~.01 ~.01 ~.1 6 ~.1 3 ~.01 ~.ω ~.09 ~.1 7 ~.05 ~.03 ~.01 ~.23 ~.1 6 ~.01 ~.1 2 ~.03 ~.04 ~.01 ~.1 0 ~.1 5 ~.ω ~.02 ~.29 ~.01 ~.01 ~.11 ~.03 ~.02 ~.ω 本 ~.02 ~.1 7 ~.1 4 ~.02 ~.21 ~.I

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