勝率と、それに相当するイロレーティング差の信頼区間計算
対局者Aと対局者Bがとある条件でベルヌーイ試行と見なせる対局を行う。対局者Aが対局で勝つ（1回のベルヌーイ試行に成功する）確率を
ここで逆に、例えば
極端に大きな勝敗数を入力を入力した際の計算精度を確保するため、 double-double 演算を一部取り入れる試みも行っています。
謝辞
このプログラムは、 レーティング推定器 (infer_rating.py) を元にして作成されました。
使い方
実行の凡例:
cargo run --release <win> <lose> [<draw> [f64|dd|both]]
<win>: 勝ち数<lose>: 負け数<draw>: 引き分け数（0.5勝換算）f64|dd|both: 計算精度f64: 倍精度浮動小数点数dd: 疑似四倍精度浮動小数点数 (double-double型)both:f64とddの両方で演算結果を出力
出力列項目:
σ: シグマ区間（相当値）x: 信頼区間に入る確率p: 信頼区間から外れる確率（片側）EloRating estimated range: イロレーティング差の信頼区間（括弧内はそれに相当する勝率）Δrange: 信頼区間の幅（イロレーティング差）error_l: 区間下側値の計算誤差指標（デバッグ用）error_u: 区間上側値の計算誤差指標（デバッグ用）
実行例
cargo run --release 120 80
---
Float Type: f64
Win-Draw-Lose: 120-0-80
Games: 200
EloRating median: +70.44
WinRate: 60.00%
DrawRate: 0.00%
σ x p | EloRating estimated range | Δrange | error_l error_u
---------------------------------+-----------------------------------------------+----------+----------------
0.000σ 0.00000000% 50.00000000% | +68.5113 ( 59.73%) ~ +72.1239 ( 60.23%) | 3.61 | +1e-14 +1e-16
0.013σ 1.00000000% 49.50000000% | +68.1978 ( 59.69%) ~ +72.4381 ( 60.28%) | 4.24 | -3e-16 -2e-15
0.025σ 2.00000000% 49.00000000% | +67.8843 ( 59.65%) ~ +72.7524 ( 60.32%) | 4.87 | +2e-16 +1e-15
0.038σ 3.00000000% 48.50000000% | +67.5707 ( 59.60%) ~ +73.0668 ( 60.36%) | 5.50 | -3e-16 +1e-15
0.050σ 4.00000000% 48.00000000% | +67.2570 ( 59.56%) ~ +73.3814 ( 60.41%) | 6.12 | +1e-14 +2e-16
0.063σ 5.00000000% 47.50000000% | +66.9432 ( 59.52%) ~ +73.6963 ( 60.45%) | 6.75 | -1e-14 +6e-17
0.075σ 6.00000000% 47.00000000% | +66.6291 ( 59.47%) ~ +74.0114 ( 60.49%) | 7.38 | -1e-14 -1e-15
0.088σ 7.00000000% 46.50000000% | +66.3148 ( 59.43%) ~ +74.3269 ( 60.54%) | 8.01 | -1e-14 -1e-14
0.100σ 8.00000000% 46.00000000% | +66.0002 ( 59.39%) ~ +74.6427 ( 60.58%) | 8.64 | -1e-14 +1e-15
0.113σ 9.00000000% 45.50000000% | +65.6852 ( 59.34%) ~ +74.9590 ( 60.62%) | 9.27 | -1e-14 -9e-15
0.126σ 10.00000000% 45.00000000% | +65.3697 ( 59.30%) ~ +75.2758 ( 60.67%) | 9.91 | -6e-17 +6e-17
0.189σ 15.00000000% 42.50000000% | +63.7847 ( 59.08%) ~ +76.8690 ( 60.89%) | 13.08 | -9e-15 +6e-17
0.253σ 20.00000000% 40.00000000% | +62.1812 ( 58.85%) ~ +78.4826 ( 61.11%) | 16.30 | -2e-15 +8e-15
0.319σ 25.00000000% 37.50000000% | +60.5521 ( 58.63%) ~ +80.1240 ( 61.33%) | 19.57 | -5e-16 -4e-16
0.385σ 30.00000000% 35.00000000% | +58.8893 ( 58.39%) ~ +81.8016 ( 61.56%) | 22.91 | +7e-16 -6e-16
0.454σ 35.00000000% 32.50000000% | +57.1836 ( 58.16%) ~ +83.5246 ( 61.79%) | 26.34 | +6e-15 +0e0
0.500σ 38.29249225% 30.85375387% | +56.0317 ( 57.99%) ~ +84.6895 ( 61.95%) | 28.66 | -2e-15 -8e-15
0.524σ 40.00000000% 30.00000000% | +55.4241 ( 57.91%) ~ +85.3044 ( 62.04%) | 29.88 | +2e-15 +3e-15
0.598σ 45.00000000% 27.50000000% | +53.5979 ( 57.65%) ~ +87.1542 ( 62.29%) | 33.56 | -4e-15 -6e-15
0.674σ 50.00000000% 25.00000000% | +51.6889 ( 57.38%) ~ +89.0906 ( 62.55%) | 37.40 | -1e-15 -9e-15
0.755σ 55.00000000% 22.50000000% | +49.6767 ( 57.10%) ~ +91.1348 ( 62.82%) | 41.46 | +4e-15 -4e-15
0.842σ 60.00000000% 20.00000000% | +47.5345 ( 56.80%) ~ +93.3145 ( 63.12%) | 45.78 | +7e-16 -1e-15
0.935σ 65.00000000% 17.50000000% | +45.2258 ( 56.47%) ~ +95.6676 ( 63.43%) | 50.44 | +4e-15 -2e-16
1.000σ 68.26894921% 15.86552539% | +43.6024 ( 56.24%) ~ +97.3249 ( 63.65%) | 53.72 | +1e-15 -6e-17
1.036σ 70.00000000% 15.00000000% | +42.6984 ( 56.11%) ~ +98.2486 ( 63.77%) | 55.55 | -6e-17 -5e-15
1.150σ 75.00000000% 12.50000000% | +39.8735 ( 55.71%) ~ +101.1393 ( 64.16%) | 61.27 | -3e-15 +3e-15
1.282σ 80.00000000% 10.00000000% | +36.6224 ( 55.25%) ~ +104.4739 ( 64.60%) | 67.85 | -1e-15 -1e-15
1.440σ 85.00000000% 7.50000000% | +32.7113 ( 54.69%) ~ +108.4966 ( 65.13%) | 75.79 | +0e0 +3e-17
1.500σ 86.63855975% 6.68072013% | +31.2152 ( 54.48%) ~ +110.0387 ( 65.33%) | 78.82 | +2e-15 -2e-15
1.645σ 90.00000000% 5.00000000% | +27.6332 ( 53.97%) ~ +113.7378 ( 65.81%) | 86.10 | +7e-17 +3e-16
1.960σ 95.00000000% 2.50000000% | +19.8502 ( 52.85%) ~ +121.8109 ( 66.85%) | 101.96 | -5e-16 -2e-17
2.000σ 95.44997361% 2.27501319% | +18.8621 ( 52.71%) ~ +122.8393 ( 66.98%) | 103.98 | +7e-17 +9e-17
2.326σ 98.00000000% 1.00000000% | +10.8140 ( 51.56%) ~ +131.2451 ( 68.04%) | 120.43 | -5e-18 -7e-17
2.500σ 98.75806693% 0.62096653% | +6.5353 ( 50.94%) ~ +135.7353 ( 68.60%) | 129.20 | +3e-17 -6e-18
2.576σ 99.00000000% 0.50000000% | +4.6676 ( 50.67%) ~ +137.7000 ( 68.84%) | 133.03 | -7e-17 +1e-16
3.000σ 99.73002039% 0.13498980% | -5.7731 ( 49.17%) ~ +148.7356 ( 70.19%) | 154.51 | +4e-17 -1e-17
3.090σ 99.80000000% 0.10000000% | -7.9930 ( 48.85%) ~ +151.0934 ( 70.47%) | 159.09 | -1e-17 +2e-17
3.291σ 99.90000000% 0.05000000% | -12.9196 ( 48.14%) ~ +156.3407 ( 71.09%) | 169.26 | +2e-17 +1e-18
3.500σ 99.95347418% 0.02326291% | -18.0709 ( 47.40%) ~ +161.8488 ( 71.74%) | 179.92 | -3e-18 -4e-18
3.719σ 99.98000000% 0.01000000% | -23.4564 ( 46.63%) ~ +167.6308 ( 72.41%) | 191.09 | +6e-19 +8e-20
3.891σ 99.99000000% 0.00500000% | -27.6754 ( 46.03%) ~ +172.1772 ( 72.93%) | 199.85 | -1e-18 +8e-19
4.000σ 99.99366575% 0.00316712% | -30.3658 ( 45.64%) ~ +175.0841 ( 73.26%) | 205.45 | -8e-19 +1e-19
4.265σ 99.99800000% 0.00100000% | -36.8809 ( 44.71%) ~ +182.1485 ( 74.05%) | 219.03 | +7e-20 -2e-20
4.417σ 99.99900000% 0.00050000% | -40.6274 ( 44.18%) ~ +186.2269 ( 74.50%) | 226.85 | -7e-20 +1e-19
4.500σ 99.99932047% 0.00033977% | -42.6656 ( 43.89%) ~ +188.4506 ( 74.74%) | 231.12 | -1e-19 -6e-20
4.753σ 99.99980000% 0.00010000% | -48.9039 ( 43.01%) ~ +195.2783 ( 75.48%) | 244.18 | -1e-20 +8e-21
4.892σ 99.99990000% 0.00005000% | -52.3079 ( 42.53%) ~ +199.0177 ( 75.87%) | 251.33 | +4e-21 +1e-20
5.000σ 99.99994267% 0.00002867% | -54.9777 ( 42.15%) ~ +201.9574 ( 76.18%) | 256.94 | -4e-22 +4e-21
5.199σ 99.99998000% 0.00001000% | -59.8914 ( 41.47%) ~ +207.3834 ( 76.74%) | 267.27 | -2e-21 -1e-21
5.327σ 99.99999000% 0.00000500% | -63.0334 ( 41.03%) ~ +210.8636 ( 77.10%) | 273.90 | +9e-22 +6e-22
5.500σ 99.99999620% 0.00000190% | -67.3098 ( 40.43%) ~ +215.6138 ( 77.58%) | 282.92 | +2e-22 -4e-22
5.612σ 99.99999800% 0.00000100% | -70.0758 ( 40.05%) ~ +218.6944 ( 77.88%) | 288.77 | +2e-22 -9e-23
5.731σ 99.99999900% 0.00000050% | -73.0094 ( 39.65%) ~ +221.9688 ( 78.21%) | 294.98 | +1e-22 +9e-23
5.998σ 99.99999980% 0.00000010% | -79.6153 ( 38.74%) ~ +229.3686 ( 78.92%) | 308.98 | -1e-23 -2e-23
6.000σ 99.99999980% 0.00000010% | -79.6696 ( 38.73%) ~ +229.4295 ( 78.93%) | 309.10 | +3e-24 -6e-25
6.109σ 99.99999990% 0.00000005% | -82.3785 ( 38.36%) ~ +232.4748 ( 79.22%) | 314.85 | -2e-23 +1e-24
cargo run --release 99 1
---
Float Type: f64
Win-Draw-Lose: 99-0-1
Games: 100
EloRating median: +798.25
WinRate: 99.00%
DrawRate: 0.00%
σ x p | EloRating estimated range | Δrange | error_l error_u
---------------------------------+-----------------------------------------------+----------+----------------
0.000σ 0.00000000% 50.00000000% | +707.7059 ( 98.33%) ~ +863.0674 ( 99.31%) | 155.36 | +2e-16 +7e-16
0.013σ 1.00000000% 49.50000000% | +706.0426 ( 98.31%) ~ +865.5879 ( 99.32%) | 159.55 | -6e-17 +0e0
0.025σ 2.00000000% 49.00000000% | +704.3840 ( 98.30%) ~ +868.1199 ( 99.33%) | 163.74 | -2e-16 +2e-16
0.038σ 3.00000000% 48.50000000% | +702.7297 ( 98.28%) ~ +870.6638 ( 99.34%) | 167.93 | -2e-16 -3e-16
0.050σ 4.00000000% 48.00000000% | +701.0796 ( 98.26%) ~ +873.2201 ( 99.35%) | 172.14 | +0e0 -1e-16
0.063σ 5.00000000% 47.50000000% | +699.4334 ( 98.25%) ~ +875.7894 ( 99.36%) | 176.36 | +6e-17 -4e-16
0.075σ 6.00000000% 47.00000000% | +697.7907 ( 98.23%) ~ +878.3721 ( 99.37%) | 180.58 | +2e-16 +6e-17
0.088σ 7.00000000% 46.50000000% | +696.1513 ( 98.21%) ~ +880.9687 ( 99.38%) | 184.82 | -6e-17 +6e-17
0.100σ 8.00000000% 46.00000000% | +694.5149 ( 98.20%) ~ +883.5797 ( 99.39%) | 189.06 | +1e-16 +3e-16
0.113σ 9.00000000% 45.50000000% | +692.8813 ( 98.18%) ~ +886.2058 ( 99.39%) | 193.32 | +6e-16 -6e-17
0.126σ 10.00000000% 45.00000000% | +691.2503 ( 98.16%) ~ +888.8474 ( 99.40%) | 197.60 | -1e-16 -2e-16
0.189σ 15.00000000% 42.50000000% | +683.1229 ( 98.08%) ~ +902.3081 ( 99.45%) | 219.19 | -8e-16 -2e-16
0.253σ 20.00000000% 40.00000000% | +675.0178 ( 97.99%) ~ +916.2470 ( 99.49%) | 241.23 | +6e-17 -2e-16
0.319σ 25.00000000% 37.50000000% | +666.9003 ( 97.89%) ~ +930.7511 ( 99.53%) | 263.85 | +3e-16 +1e-16
0.385σ 30.00000000% 35.00000000% | +658.7341 ( 97.79%) ~ +945.9223 ( 99.57%) | 287.19 | -1e-16 -2e-16
0.454σ 35.00000000% 32.50000000% | +650.4793 ( 97.69%) ~ +961.8827 ( 99.61%) | 311.40 | -3e-16 +0e0
0.500σ 38.29249225% 30.85375387% | +644.9739 ( 97.62%) ~ +972.8941 ( 99.63%) | 327.92 | -4e-16 +2e-16
0.524σ 40.00000000% 30.00000000% | +642.0914 ( 97.58%) ~ +978.7811 ( 99.64%) | 336.69 | +0e0 +3e-16
0.598σ 45.00000000% 27.50000000% | +633.5184 ( 97.46%) ~ +996.8030 ( 99.68%) | 363.28 | -4e-16 -2e-16
0.674σ 50.00000000% 25.00000000% | +624.6982 ( 97.33%) ~ +1016.1849 ( 99.71%) | 391.49 | -1e-16 +2e-16
0.755σ 55.00000000% 22.50000000% | +615.5537 ( 97.19%) ~ +1037.2359 ( 99.75%) | 421.68 | +3e-16 -6e-17
0.842σ 60.00000000% 20.00000000% | +605.9857 ( 97.04%) ~ +1060.3724 ( 99.78%) | 454.39 | -3e-15 -1e-16
0.935σ 65.00000000% 17.50000000% | +595.8615 ( 96.86%) ~ +1086.1762 ( 99.81%) | 490.31 | -1e-15 -3e-17
1.000σ 68.26894921% 15.86552539% | +588.8556 ( 96.74%) ~ +1104.8789 ( 99.83%) | 516.02 | +3e-17 -6e-17
1.036σ 70.00000000% 15.00000000% | +584.9944 ( 96.67%) ~ +1115.4972 ( 99.84%) | 530.50 | -1e-16 +3e-17
1.150σ 75.00000000% 12.50000000% | +573.1064 ( 96.44%) ~ +1149.6507 ( 99.87%) | 576.54 | -3e-17 +3e-17
1.282σ 80.00000000% 10.00000000% | +559.7501 ( 96.17%) ~ +1190.8373 ( 99.89%) | 631.09 | -6e-16 +1e-17
1.440σ 85.00000000% 7.50000000% | +544.1226 ( 95.82%) ~ +1243.1801 ( 99.92%) | 699.06 | +7e-17 -4e-17
1.500σ 86.63855975% 6.68072013% | +538.2672 ( 95.68%) ~ +1264.0395 ( 99.93%) | 725.77 | +3e-16 +3e-17
1.645σ 90.00000000% 5.00000000% | +524.5129 ( 95.34%) ~ +1315.9312 ( 99.95%) | 791.42 | -2e-17 -2e-17
1.960σ 95.00000000% 2.50000000% | +495.8430 ( 94.55%) ~ +1438.6076 ( 99.97%) | 942.76 | +8e-17 -1e-17
2.000σ 95.44997361% 2.27501319% | +492.3162 ( 94.45%) ~ +1455.1914 ( 99.98%) | 962.88 | -1e-17 -3e-18
2.326σ 98.00000000% 1.00000000% | +464.4706 ( 93.55%) ~ +1599.1190 ( 99.99%) | 1134.65 | -5e-18 +3e-18
2.500σ 98.75806693% 0.62096653% | +450.2708 ( 93.03%) ~ +1682.2265 ( 99.99%) | 1231.96 | +1e-17 -6e-18
2.576σ 99.00000000% 0.50000000% | +444.1962 ( 92.80%) ~ +1719.9724 ( 99.99%) | 1275.78 | +3e-17 +2e-18
3.000σ 99.73002039% 0.13498980% | +411.5310 ( 91.44%) ~ +1947.7611 (100.00%) | 1536.23 | +2e-18 +2e-18
3.090σ 99.80000000% 0.10000000% | +404.8523 ( 91.14%) ~ +1999.9122 (100.00%) | 1595.06 | -3e-18 +2e-19
3.291σ 99.90000000% 0.05000000% | +390.3408 ( 90.44%) ~ +2120.3681 (100.00%) | 1730.03 | -4e-19 +1e-19
3.500σ 99.95347418% 0.02326291% | +375.6034 ( 89.68%) ~ +2253.3140 (100.00%) | 1877.71 | -3e-19 +1e-19
3.719σ 99.98000000% 0.01000000% | +360.6447 ( 88.86%) ~ +2399.9912 (100.00%) | 2039.35 | +8e-20 +8e-20
3.891σ 99.99000000% 0.00500000% | +349.2284 ( 88.19%) ~ +2520.4076 (100.00%) | 2171.18 | +4e-20 +4e-20
4.000σ 99.99366575% 0.00316712% | +342.0804 ( 87.75%) ~ +2599.7312 (100.00%) | 2257.65 | -2e-20 -2e-20
4.265σ 99.99800000% 0.00100000% | +325.1768 ( 86.67%) ~ +2799.9991 (100.00%) | 2474.82 | +2e-20 -3e-21
4.417σ 99.99900000% 0.00050000% | +315.7040 ( 86.02%) ~ +2920.4116 (100.00%) | 2604.71 | -2e-21 -2e-21
4.500σ 99.99932047% 0.00033977% | +310.6232 ( 85.67%) ~ +2987.5271 (100.00%) | 2676.90 | -8e-21 -3e-21
4.753σ 99.99980000% 0.00010000% | +295.3757 ( 84.56%) ~ +3199.9999 (100.00%) | 2904.62 | -1e-21 +4e-22
4.892σ 99.99990000% 0.00005000% | +287.2410 ( 83.94%) ~ +3320.4120 (100.00%) | 3033.17 | -6e-22 +2e-22
5.000σ 99.99994267% 0.00002867% | +280.9484 ( 83.44%) ~ +3417.0582 (100.00%) | 3136.11 | +2e-21 +2e-22
5.199σ 99.99998000% 0.00001000% | +269.5602 ( 82.52%) ~ +3600.0000 (100.00%) | 3330.44 | +3e-22 -5e-23
5.327σ 99.99999000% 0.00000500% | +262.4047 ( 81.91%) ~ +3720.4120 (100.00%) | 3458.01 | +5e-23 +5e-23
5.500σ 99.99999620% 0.00000190% | +252.8173 ( 81.08%) ~ +3888.5940 (100.00%) | 3635.78 | -2e-23 -2e-23
5.612σ 99.99999800% 0.00000100% | +246.7063 ( 80.54%) ~ +4000.0000 (100.00%) | 3753.29 | -2e-23 -2e-23
5.731σ 99.99999900% 0.00000050% | +240.2994 ( 79.95%) ~ +4120.4120 (100.00%) | 3880.11 | +0e0 +0e0
5.998σ 99.99999980% 0.00000010% | +226.1423 ( 78.61%) ~ +4400.0000 (100.00%) | 4173.86 | +6e-25 +6e-25
6.000σ 99.99999980% 0.00000010% | +226.0275 ( 78.60%) ~ +4402.3457 (100.00%) | 4176.32 | +6e-24 -6e-25
6.109σ 99.99999990% 0.00000005% | +220.3266 ( 78.05%) ~ +4520.4120 (100.00%) | 4300.09 | -6e-25 -6e-25
cargo run --release 987654321098765 123456789012345 0 dd
---
Float Type: ddreal::DDReal
Win-Draw-Lose: 987654321098765-0-123456789012345
Games: 1111111110111110
EloRating median: +361.24
WinRate: 88.89%
DrawRate: 0.00%
σ x p | EloRating estimated range | Δrange | error_l error_u
---------------------------------+-----------------------------------------------+----------+----------------
0.000σ 0.00000000% 50.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | -0.00 | +2e-1 +2e-1
0.013σ 1.00000000% 49.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -2e-1 -5e-1
0.038σ 3.00000000% 48.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | -0.00 | +5e-2 -3e-1
0.050σ 4.00000000% 48.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -2e-2 -1e-2
0.063σ 5.00000000% 47.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | -0.00 | -4e-3 +2e-1
0.075σ 6.00000000% 47.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -1e-17 -8e-18
0.088σ 7.00000000% 46.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -5e-18 -6e-18
0.100σ 8.00000000% 46.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -5e-18 -2e-18
0.113σ 9.00000000% 45.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -6e-18 +2e-18
0.126σ 10.00000000% 45.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +6e-26 +3e-18
0.189σ 15.00000000% 42.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -1e-18 -8e-18
0.253σ 20.00000000% 40.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +2e-18 +9e-23
0.319σ 25.00000000% 37.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +2e-18 -9e-19
0.385σ 30.00000000% 35.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -2e-22 -5e-19
0.454σ 35.00000000% 32.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +2e-19 -3e-18
0.500σ 38.29249225% 30.85375387% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +2e-18 -2e-18
0.524σ 40.00000000% 30.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -3e-18 +3e-18
0.598σ 45.00000000% 27.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +5e-18 -2e-18
0.674σ 50.00000000% 25.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -7e-19 -1e-18
0.755σ 55.00000000% 22.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +2e-18 +3e-18
0.842σ 60.00000000% 20.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +1e-18 -2e-18
0.935σ 65.00000000% 17.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -7e-19 -4e-19
1.000σ 68.26894921% 15.86552539% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -1e-18 -6e-19
1.036σ 70.00000000% 15.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +2e-19 +1e-18
1.150σ 75.00000000% 12.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +3e-19 -5e-19
1.282σ 80.00000000% 10.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -6e-19 -5e-19
1.440σ 85.00000000% 7.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -1e-19 -3e-22
1.500σ 86.63855975% 6.68072013% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +7e-19 -5e-19
1.645σ 90.00000000% 5.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +5e-19 -4e-19
1.960σ 95.00000000% 2.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -4e-20 +3e-20
2.000σ 95.44997361% 2.27501319% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +9e-22 -8e-20
2.326σ 98.00000000% 1.00000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +4e-20 +2e-19
2.500σ 98.75806693% 0.62096653% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -8e-20 +6e-20
2.576σ 99.00000000% 0.50000000% | +361.2360 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -4e-20 -7e-21
3.000σ 99.73002039% 0.13498980% | +361.2359 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | -2e-20 -2e-20
3.090σ 99.80000000% 0.10000000% | +361.2359 ( 88.89%) ~ +361.2360 ( 88.89%) | 0.00 | +5e-21 -2e-21
3.291σ 99.90000000% 0.05000000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -5e-21 +3e-21
3.500σ 99.95347418% 0.02326291% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -4e-21 -1e-21
3.719σ 99.98000000% 0.01000000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +9e-22 +6e-22
3.891σ 99.99000000% 0.00500000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -5e-22 -1e-22
4.000σ 99.99366575% 0.00316712% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -4e-23 +9e-24
4.265σ 99.99800000% 0.00100000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +9e-24 -9e-23
4.417σ 99.99900000% 0.00050000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -1e-23 -4e-23
4.500σ 99.99932047% 0.00033977% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +2e-23 +2e-23
4.753σ 99.99980000% 0.00010000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +2e-24 -8e-24
4.892σ 99.99990000% 0.00005000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +3e-24 +2e-24
5.000σ 99.99994267% 0.00002867% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +9e-25 +9e-25
5.199σ 99.99998000% 0.00001000% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +6e-26 -2e-25
5.327σ 99.99999000% 0.00000500% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -4e-25 +5e-25
5.500σ 99.99999620% 0.00000190% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -2e-26 -2e-25
5.612σ 99.99999800% 0.00000100% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +1e-26 +8e-26
5.731σ 99.99999900% 0.00000050% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +3e-26 -5e-27
5.998σ 99.99999980% 0.00000010% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | -7e-27 -1e-27
6.000σ 99.99999980% 0.00000010% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +2e-26 +6e-27
6.109σ 99.99999990% 0.00000005% | +361.2359 ( 88.89%) ~ +361.2361 ( 88.89%) | 0.00 | +7e-28 +4e-27
数学的な話
正則化不完全ベータ関数と二項分布の累積分布関数との関連
正則化不完全ベータ関数は、ベータ分布の累積分布関数であり、二項分布の確率変数
$X$ の累積分布関数$F(k;,n,p)=\Pr\left(X\le k\right)$ に関連している。ここで、1回のベルヌーイ試行で成功する確率を$p$ 、ベルヌーイ試行する回数を$n$ とする:
$F(k;,n,p)=\Pr\left(X\le k\right)=I_{1-p}(n-k,k+1)=1-I_p(k+1,n-k).$
ベルヌーイ試行
確率論や統計学において、ベルヌーイ試行（ベルヌーイしこう、英語: Bernoulli trial）または二項試行（にこうしこう、英語: binomial trial）とは、取り得る結果が「成功」「失敗」の2つのみであり、各試行において成功の確率が同じであるランダム試行である。この名前は、17世紀のスイスの数学者であるヤコブ・ベルヌーイにちなんで名付けられた。ベルヌーイは、1713年の著書『推測法』(Ars Conjectandi)でこの試行を分析した。
イロレーティング
もっと数学的な話
特殊関数 開発メモ の方もご覧ください。