v1.0 2017 12 28 1 125 (11 3 ) (4 29 ) (12 23 ) 125 125 1 N = 365 M = 125 P (M, N n) n 2 7 (1) 7 1 (1+6) 1 1 6 1 (2+5) 2 1 5 1 (1+2+2+2) 1 1 2 1 2 1 2 1 15 (1+2+2+2) 4 1 1
N C 1 7 C 1 : 1 1 : 1 1 N 1 C 3 : (2+2+2) 3 6 C 2 3 6 2 3 (2 3)!/(2!) 3 6 1 1,833,153,121,800 1 1 7 7 N M ( 365 7 ) #holidays 1 2 3 4 5 6 7 partitions (7) (1+6) (1+1+5) (1+1+1+4) (1+1+1+1+3) (1+1+1+1+1+2) (1+1+1+1+1+1+1) (2+5) (1+2+4) (1+1+2+3) (1+1+1+2+2) (3+4) (1+3+3) (1+2+2+2) (2+2+3) Partition Number ( ) = 15 #cases 365 837018 1.45E+10 6.11E+12 8.82E+14 4.76E+16 8.15E+17 1 / 365 7 Probability 1 #holidays 6 1: 7 7 365 7 2 M = 125 10 9 M = 90 5.7 10 7 100 M = 125 80 7 2
partition number 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 0 20 40 60 80 100 N 2: N (partition number) 3 (Monte Carlo:MC) M 3 10 5 (M = 125) 106 (Root Mean Square: RMS) 3.5 106 [1] 1 12 2.2 RMS 30 ( 31 ) 4 105.98 ± 0.01 105.95 ± 0.01 3
Number of holidays 25000 20000 15000 10000 5000 M=50 Mean:46.7889 M=125 RMS :1.63208 M=200 Mean:105.984 M=300 RMS :3.48609 M=500 Mean:154.165 RMS :4.71311 Mean:204.737 RMS :5.67526 Mean:272.421 RMS :6.06256 0 0 50 100 150 200 250 300 350 3: (M = 50) (M = 125) (M = 200) (M = 300) (M = 500) 1: ( 12 ) [1] % (RMS) 1 8.35 0.09 2 8.26-0.36 3 8.17-0.81 4 8.08-1.26 5 8.26-0.36 6 8.17-0.81 7 8.52 0.94 8 8.61 1.39 9 8.79 2.29 10 8.35 0.09 11 8.26-0.36 12 8.17-0.81 8.33 ± 0.20 (RMS) 4
Number of holidays 25000 20000 15000 10000 5000 M=50 Mean:46.7837 M=125 RMS :1.63511 M=200 Mean:105.953 M=300 RMS :3.4784 M=500 Mean:154.141 RMS :4.70842 Mean:204.701 RMS :5.6498 Mean:272.409 RMS :6.0882 0 0 50 100 150 200 250 300 350 4: (M = 50) (M = 125) (M = 200) (M = 300) (M = 500) 4 4.1 Q(M n) n M M n n M M 1 2 Q(M n) = n M n C 1 Q(M 1) n C 2 Q(M 2) (1) n 1 M min(m,n 1) Q(M n) = n M nc i Q(M i) (2) Q(M n) N n M i=1 P (M, N n) = N C n Q(M n) N M (3) (2) Q(M i) i = 1 M P (M, N n) 5
4.2 Q(M i) N M = 365 125 10 320 Python P (M, N n) ( ) Python O(10 308 ) 10 100 ( ) 5 P (125, 365 n) 106.0 RMS 3.5 3 0.12 0.10 #emperos=125 average=105.97 days in 365 days sigma= 3.48 days 0.08 PDF 0.06 0.04 0.02 0.00 0 20 40 60 80 100 120 #holidays 5: 5 1 P holiday N P holiday 1 P holiday 5.1 M N ( ) ( ) 1. α 1/N 1 α. 6
2: Table of birthdays. : a brithday, : not a birthday. Day 0 1 N-1 Person 0 1 M-1 2. 1 ( 1 ) 1 P holiday P workday 1 P holiday 5.2 M N [α + (1 α)] M+N = 1 (4) ( ) 0 2 P workday P holiday = P workday 1 5.3 1 N C 1 α 1 (1 α) N 1 (5) M [ NC 1 α 1 (1 α) N 1] M (6) 5.4 1 1 (1 α) N 1 C 1 α 1 (1 α) N 2 (7) M [ (1 α) N 1 C 1 α 1 (1 α) N 2] M (8) 7
5.5 [ (1 α) N 1 C 1 α 1 (1 α) N 2] M P workday = [ N C 1 α 1 (1 α) N 1 ] M (9) [ (1 α) (N 1)α 1 (1 α) N 2 ] M = Nα 1 (1 α) N 1 (10) = [1 1/N] M (11) =(1 α) M (12) P holiday =1 P workday = 1 (1 α) M = 1 (1 1/N) M (13) P workday = (1 α) M 0 2 M 5.6 M = 125, N = 365 P holiday =1 (1 1/365) 125 = 0.29 (14) µ µ =N P holiday = 106( ). (15) 5.7 P holiday N 2 RMS RMS = N P holiday (1 P holiday ) = 8.7( ). (16) MC 2 P holiday 1 5.8 2 1. 1 ( 1 ) N N M 1 P workday =(1 1/N) M (17) 8
0 350 365.*(1-(1-1./365.)**x) 10 sqrt(365.*(1-(1-1./365.)**x)*(1-1./365.)**x) 9 300 8 250 7 Mean number of holidays 200 150 RMS of holidays 6 5 4 100 3 2 50 1 0 200 400 600 800 1000 0 0 200 400 600 800 1000 Number of persons Number of persons 6: 2 7: 2 RMS 3: Table of birthdays. : a brithday. Day 0 1 N-1 Person 0 1 M-1 6 3 4 5 Python 2.7.13 (64bit ) 1 M 1000 12 n NC n Q(M n)/n M = N [ 1 (1 1/N) M] (18) n 3 8 1 M 1000 RMS 9 ( ) 9
(a) formula MC (b) 8: (a): ( ) ( ) M ( ) (b): (RMS/ 10 5 ) formula MC + simple 9: RMS M ( ) ( ) ( ) ( ) 10
7 125 [2] [3, 4] AKB48 138 114 (3) 115.0 RMS 3.7 NMB48 SKE48 HKT48 [4] [5] [6] 10 11 46 43 12,102 12,102 366 365.0 ± 0.0 10: 8 125 106.0 ± 3.5 n = N [ 1 ( 1 1 ) ] M N (19) (20) 11
jurist 5 moviedir n - nexpected 0 5 hello_pro NMB48 HKT48 AKB48 Nogizaka46 SKE48 kyoutei figurescating sumo medicineboxing baseball basketball politicians prowrestling golf volleyball Takarazuka actors 10 translators tennis 15 jockey 10 2 10 3 10 4 M 11: n = n n NC n Q(M n)/n M (21) (RMS) M MPPC [1] [accessed 2017 Dec. 26]. http://www.mhlw.go.jp/toukei/saikin/hw/jinkou/tokusyu/ syussyo-4/syussyo1-2.html. [2] Wikipedia [accessed 2017 Dec. 26]. https://ja.wikipedia.org/wiki/. [3] AKB [accessed 2017 Dec. 26]. https://www.akb48.co.jp. 12
[4] [accessed 2017 Dec. 26]. http://oshitan.com. [5] [accessed 2017 Dec. 26]. http://zukaholic.3album.net/student/ index/s_birthday/. [6] ( ) [accessed 2017 Dec. 26]. https://www.tanjoubi.org/birth/. 13