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製錬成功率がpのとき、製錬が成功するまでの回数Xは幾何分布に従い、期待値は E[X] = 1/p で分散は V[X] = (1-p)/p²
リペアパウダーの必要回数の平均値は E[X] -1
標準偏差σ は、分散の平方根 σ = √V[X] = (1/p)√(1-p)
f(t) = P(X≦E[X]+tσ) とし、p,f(t)を%表示で表すと以下の表のようになる。
| p | E[X] | σ | f(0) | f(1) | f(2) | p | E[X] | σ | f(0) | f(1) | f(2) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 100.00 | 99.50 | 63.40 | 86.47 | 95.00 | 21 | 4.76 | 4.23 | 61.05 | 84.83 | 95.33 | |
| 2 | 50.00 | 49.50 | 63.58 | 86.47 | 94.97 | 22 | 4.55 | 4.01 | 62.98 | 86.30 | 94.93 | |
| 3 | 33.33 | 32.83 | 63.40 | 86.61 | 94.95 | 23 | 4.35 | 3.82 | 64.85 | 87.64 | 94.36 | |
| 4 | 25.00 | 24.49 | 63.96 | 86.47 | 94.92 | 24 | 4.17 | 3.63 | 66.64 | 85.35 | 95.11 | |
| 5 | 20.00 | 19.49 | 64.15 | 86.47 | 94.90 | 25 | 4.00 | 3.46 | 68.36 | 86.65 | 94.37 | |
| 6 | 16.67 | 16.16 | 62.84 | 86.19 | 94.87 | 26 | 3.85 | 3.31 | 59.48 | 87.85 | 95.08 | |
| 7 | 14.29 | 13.78 | 63.80 | 86.89 | 94.90 | 27 | 3.70 | 3.16 | 61.10 | 84.87 | 95.70 | |
| 8 | 12.50 | 11.99 | 63.23 | 86.48 | 95.03 | 28 | 3.57 | 3.03 | 62.68 | 86.07 | 94.80 | |
| 9 | 11.11 | 10.60 | 64.56 | 86.20 | 95.11 | 29 | 3.45 | 2.91 | 64.21 | 87.19 | 95.42 | |
| 10 | 10.00 | 9.49 | 65.13 | 86.49 | 94.77 | 30 | 3.33 | 2.79 | 65.70 | 88.24 | 94.24 | |
| 11 | 9.09 | 8.58 | 64.96 | 86.21 | 95.17 | 31 | 3.23 | 2.68 | 67.15 | 84.36 | 94.86 | |
| 12 | 8.33 | 7.82 | 64.04 | 87.07 | 94.71 | 32 | 3.13 | 2.58 | 68.56 | 85.46 | 95.43 | |
| 13 | 7.69 | 7.17 | 62.27 | 85.77 | 95.33 | 33 | 3.03 | 2.48 | 69.92 | 86.50 | 93.94 | |
| 14 | 7.14 | 6.62 | 65.21 | 85.92 | 95.10 | 34 | 2.94 | 2.39 | 56.44 | 87.48 | 94.54 | |
| 15 | 6.67 | 6.15 | 62.29 | 85.78 | 94.64 | 35 | 2.86 | 2.30 | 57.75 | 88.40 | 95.10 | |
| 16 | 6.25 | 5.73 | 64.87 | 85.31 | 94.84 | 36 | 2.78 | 2.22 | 59.04 | 89.26 | 95.60 | |
| 17 | 5.88 | 5.36 | 60.61 | 87.12 | 94.93 | 37 | 2.70 | 2.15 | 60.31 | 84.25 | 93.75 | |
| 18 | 5.56 | 5.03 | 62.93 | 86.26 | 94.90 | 38 | 2.63 | 2.07 | 61.56 | 85.22 | 94.32 | |
| 19 | 5.26 | 4.74 | 65.13 | 87.84 | 94.77 | 39 | 2.56 | 2.00 | 62.79 | 86.15 | 94.85 | |
| 20 | 5.00 | 4.47 | 67.23 | 86.58 | 94.50 | 40 | 2.50 | 1.94 | 64.00 | 87.04 | 95.33 |
| p | E[X] | σ | f(0) | f(1) | f(2) | p | E[X] | σ | f(0) | f(1) | f(2) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 41 | 2.44 | 1.87 | 65.19 | 87.88 | 95.78 | 61 | 1.64 | 1.02 | 61.00 | 84.79 | 94.07 | |
| 42 | 2.38 | 1.81 | 66.36 | 88.68 | 96.19 | 62 | 1.61 | 0.99 | 62.00 | 85.56 | 94.51 | |
| 43 | 2.33 | 1.76 | 67.51 | 89.44 | 93.98 | 63 | 1.59 | 0.97 | 63.00 | 86.31 | 94.93 | |
| 44 | 2.27 | 1.70 | 68.64 | 82.44 | 94.49 | 64 | 1.56 | 0.94 | 64.00 | 87.04 | 95.33 | |
| 45 | 2.22 | 1.65 | 69.75 | 83.36 | 94.97 | 65 | 1.54 | 0.91 | 65.00 | 87.75 | 95.71 | |
| 46 | 2.17 | 1.60 | 70.84 | 84.25 | 95.41 | 66 | 1.52 | 0.88 | 66.00 | 88.44 | 96.07 | |
| 47 | 2.13 | 1.55 | 71.91 | 85.11 | 95.82 | 67 | 1.49 | 0.86 | 67.00 | 89.11 | 96.41 | |
| 48 | 2.08 | 1.50 | 72.96 | 85.94 | 96.20 | 68 | 1.47 | 0.83 | 68.00 | 89.76 | 96.72 | |
| 49 | 2.04 | 1.46 | 73.99 | 86.73 | 93.23 | 69 | 1.45 | 0.81 | 69.00 | 90.39 | 97.02 | |
| 50 | 2.00 | 1.41 | 75.00 | 87.50 | 93.75 | 70 | 1.43 | 0.78 | 70.00 | 91.00 | 91.00 | |
| 51 | 1.96 | 1.37 | 51.00 | 88.24 | 94.24 | 71 | 1.41 | 0.76 | 71.00 | 91.59 | 91.59 | |
| 52 | 1.92 | 1.33 | 52.00 | 88.94 | 94.69 | 72 | 1.39 | 0.73 | 72.00 | 92.16 | 92.16 | |
| 53 | 1.89 | 1.29 | 53.00 | 89.62 | 95.12 | 73 | 1.37 | 0.71 | 73.00 | 92.71 | 92.71 | |
| 54 | 1.85 | 1.26 | 54.00 | 90.27 | 95.52 | 74 | 1.35 | 0.69 | 74.00 | 93.24 | 93.24 | |
| 55 | 1.82 | 1.22 | 55.00 | 90.89 | 95.90 | 75 | 1.33 | 0.67 | 75.00 | 93.75 | 93.75 | |
| 56 | 1.79 | 1.18 | 56.00 | 80.64 | 96.25 | 76 | 1.32 | 0.64 | 76.00 | 76.00 | 94.24 | |
| 57 | 1.75 | 1.15 | 57.00 | 81.51 | 96.58 | 77 | 1.30 | 0.62 | 77.00 | 77.00 | 94.71 | |
| 58 | 1.72 | 1.12 | 58.00 | 82.36 | 92.59 | 78 | 1.28 | 0.60 | 78.00 | 78.00 | 95.16 | |
| 59 | 1.69 | 1.09 | 59.00 | 83.19 | 93.11 | 79 | 1.27 | 0.58 | 79.00 | 79.00 | 95.59 | |
| 60 | 1.67 | 1.05 | 60.00 | 84.00 | 93.60 | 80 | 1.25 | 0.56 | 80.00 | 80.00 | 96.00 |
| p | E[X] | σ | f(0) | f(1) | f(2) | p | E[X] | σ | f(0) | f(1) | f(2) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81 | 1.23 | 0.54 | 81.00 | 81.00 | 96.39 | 91 | 1.10 | 0.33 | 91.00 | 91.00 | 91.00 | |
| 82 | 1.22 | 0.52 | 82.00 | 82.00 | 96.76 | 92 | 1.09 | 0.31 | 92.00 | 92.00 | 92.00 | |
| 83 | 1.20 | 0.50 | 83.00 | 83.00 | 97.11 | 93 | 1.08 | 0.28 | 93.00 | 93.00 | 93.00 | |
| 84 | 1.19 | 0.48 | 84.00 | 84.00 | 97.44 | 94 | 1.06 | 0.26 | 94.00 | 94.00 | 94.00 | |
| 85 | 1.18 | 0.46 | 85.00 | 85.00 | 97.75 | 95 | 1.05 | 0.24 | 95.00 | 95.00 | 95.00 | |
| 86 | 1.16 | 0.44 | 86.00 | 86.00 | 98.04 | 96 | 1.04 | 0.21 | 96.00 | 96.00 | 96.00 | |
| 87 | 1.15 | 0.41 | 87.00 | 87.00 | 87.00 | 97 | 1.03 | 0.18 | 97.00 | 97.00 | 97.00 | |
| 88 | 1.14 | 0.39 | 88.00 | 88.00 | 88.00 | 98 | 1.02 | 0.14 | 98.00 | 98.00 | 98.00 | |
| 89 | 1.12 | 0.37 | 89.00 | 89.00 | 89.00 | 99 | 1.01 | 0.10 | 99.00 | 99.00 | 99.00 | |
| 90 | 1.11 | 0.35 | 90.00 | 90.00 | 90.00 | 100 | 1.00 | 0.00 | 100.00 | 100.00 | 100.00 |
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