Methods of Calculating
Probabilities for a single die can be calculated directly.
Subsequent probabilities were calculated using a spreadsheet program, using the following formulae:
P(at least S successes on n dice)=P(at least S successes on n-1 dice)+sum_{s=0}^{S-1} P(at least s successes on n-1 dice)P(exactly S-s successes on 1 die)
Some simple formulas
A few things are simple enough to calculate on the fly.
If all rolls of r or better are rerolled, then the average number of successes obtained when rolling n dice is 3n/(r-1). (In particular, if 10s and only 10s are rerolled, then the average is 3n/(10-1)=n/3, and if no numbers are rerolled, then the average is 3n/(11-1)=0.3n. When using formulas, the easiest way to describe the no-rerolls situation is to call it "11-again" rather than "none-again".)
If failures are rerolled once ("rote actions"), then the average number of successes is 5.1n/(r-1) (usually 0.51n, under 10-again).
For non-rote actions, the probability of rolling no successes on n dice is 0.7^n, and so the probability of rolling at least one success is 1-0.7^n. For rote actions, failures may be rerolled once, and so the probability of rolling at least one success is 1-0.7^(2n).
With one die, the probability of rolling at least 1 success is 0.51 (rote actions) or 0.3 (anything else).
The probability of rolling at least S successes on one die under r-again, for S>1, is (11-r)/3 * 0.3 * P(at least S-1 successes) = 0.3*(1.1-r/10)^(S-1) (for non-rote actions) or 0.51*(1.1-r/10)^(S-1) (for rote actions).
10 again
This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the normal Chronicles of Darkness rules.
| Successes | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 die | 30.00 | 3.00 | 0.30 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 2 dice | 51.00 | 13.20 | 2.13 | 0.29 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 3 dice | 65.70 | 26.01 | 6.73 | 1.30 | 0.21 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 4 dice | 75.99 | 38.94 | 13.80 | 3.64 | 0.77 | 0.14 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 5 dice | 83.19 | 50.77 | 22.53 | 7.56 | 2.02 | 0.45 | 0.09 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
| 6 dice | 88.23 | 61.00 | 32.03 | 13.00 | 4.23 | 1.14 | 0.27 | 0.05 | 0.01 | 0.00 | 0.00 | 0.00 |
| 7 dice | 91.76 | 69.52 | 41.57 | 19.66 | 7.53 | 2.40 | 0.65 | 0.15 | 0.03 | 0.00 | 0.00 | 0.00 |
| 8 dice | 94.23 | 76.44 | 50.65 | 27.14 | 11.91 | 4.38 | 1.38 | 0.38 | 0.09 | 0.02 | 0.00 | 0.00 |
| 9 dice | 95.96 | 81.95 | 58.94 | 35.02 | 17.27 | 7.17 | 2.56 | 0.79 | 0.22 | 0.05 | 0.01 | 0.00 |
| 10 dice | 97.17 | 86.27 | 66.27 | 42.93 | 23.39 | 10.81 | 4.30 | 1.50 | 0.46 | 0.13 | 0.03 | 0.00 |
| 11 dice | 98.02 | 89.63 | 72.61 | 50.57 | 30.01 | 15.25 | 6.70 | 2.58 | 0.88 | 0.27 | 0.07 | 0.01 |
| 12 dice | 98.61 | 92.20 | 77.97 | 57.72 | 36.89 | 20.37 | 9.78 | 4.12 | 1.54 | 0.52 | 0.16 | 0.04 |
| 13 dice | 99.03 | 94.17 | 82.44 | 64.24 | 43.79 | 26.01 | 13.52 | 6.19 | 2.53 | 0.92 | 0.31 | 0.09 |
| 14 dice | 99.32 | 95.65 | 86.11 | 70.07 | 50.51 | 32.02 | 17.87 | 8.83 | 3.89 | 1.54 | 0.55 | 0.18 |
| 15 dice | 99.52 | 96.77 | 89.08 | 75.18 | 56.89 | 38.20 | 22.73 | 12.03 | 5.69 | 2.43 | 0.94 | 0.33 |
9 again
This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the 9-again system.
| Successes | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 die | 30.00 | 6.00 | 1.20 | 0.24 | 0.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 2 dice | 51.00 | 17.40 | 4.92 | 1.27 | 0.31 | 0.07 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 3 dice | 65.70 | 30.42 | 11.26 | 3.63 | 1.07 | 0.29 | 0.07 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 |
| 4 dice | 75.99 | 43.06 | 19.54 | 7.58 | 2.63 | 0.84 | 0.25 | 0.07 | 0.02 | 0.00 | 0.00 | 0.00 |
| 5 dice | 83.19 | 54.38 | 28.86 | 13.03 | 5.20 | 1.89 | 0.64 | 0.20 | 0.06 | 0.01 | 0.00 | 0.00 |
| 6 dice | 88.23 | 64.03 | 38.44 | 19.69 | 8.88 | 3.62 | 1.36 | 0.47 | 0.15 | 0.05 | 0.01 | 0.00 |
| 7 dice | 91.76 | 71.99 | 47.71 | 27.17 | 13.62 | 6.15 | 2.54 | 0.98 | 0.35 | 0.12 | 0.04 | 0.01 |
| 8 dice | 94.23 | 78.42 | 56.28 | 35.05 | 19.26 | 9.52 | 4.30 | 1.80 | 0.70 | 0.26 | 0.09 | 0.03 |
| 9 dice | 95.96 | 83.51 | 63.94 | 42.95 | 25.58 | 13.70 | 6.70 | 3.03 | 1.28 | 0.51 | 0.19 | 0.07 |
| 10 dice | 97.17 | 87.49 | 70.61 | 50.58 | 32.32 | 18.61 | 9.78 | 4.75 | 2.15 | 0.91 | 0.37 | 0.14 |
| 11 dice | 98.02 | 90.56 | 76.28 | 57.72 | 39.22 | 24.11 | 13.53 | 7.01 | 3.38 | 1.53 | 0.65 | 0.26 |
| 12 dice | 98.61 | 92.92 | 81.04 | 64.24 | 46.08 | 30.01 | 17.88 | 9.83 | 5.03 | 2.41 | 1.09 | 0.47 |
| 13 dice | 99.03 | 94.71 | 84.96 | 70.07 | 52.69 | 36.15 | 22.75 | 13.22 | 7.15 | 3.62 | 1.73 | 0.78 |
| 14 dice | 99.32 | 96.06 | 88.16 | 75.17 | 58.93 | 42.36 | 28.01 | 17.13 | 9.75 | 5.20 | 2.61 | 1.24 |
| 15 dice | 99.52 | 97.08 | 90.73 | 79.58 | 64.68 | 48.48 | 33.54 | 21.50 | 12.84 | 7.19 | 3.79 | 1.89 |
8 again
This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the 8-again system.
| Successes | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 die | 30.00 | 9.00 | 2.70 | 0.81 | 0.24 | 0.07 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 2 dice | 51.00 | 21.60 | 8.37 | 3.07 | 1.09 | 0.37 | 0.12 | 0.04 | 0.01 | 0.00 | 0.00 | 0.00 |
| 3 dice | 65.70 | 34.83 | 16.30 | 7.04 | 2.87 | 1.12 | 0.42 | 0.15 | 0.05 | 0.02 | 0.00 | 0.00 |
| 4 dice | 75.99 | 47.17 | 25.56 | 12.60 | 5.79 | 2.52 | 1.05 | 0.42 | 0.16 | 0.06 | 0.02 | 0.00 |
| 5 dice | 83.19 | 57.98 | 35.29 | 19.41 | 9.88 | 4.73 | 2.16 | 0.94 | 0.40 | 0.16 | 0.06 | 0.02 |
| 6 dice | 88.23 | 67.05 | 44.82 | 27.03 | 15.02 | 7.82 | 3.86 | 1.82 | 0.82 | 0.36 | 0.15 | 0.06 |
| 7 dice | 91.76 | 74.47 | 53.71 | 35.03 | 21.03 | 11.78 | 6.23 | 3.14 | 1.52 | 0.71 | 0.32 | 0.14 |
| 8 dice | 94.23 | 80.39 | 61.72 | 43.04 | 27.63 | 16.53 | 9.32 | 5.00 | 2.56 | 1.26 | 0.60 | 0.28 |
| 9 dice | 95.96 | 85.06 | 68.72 | 50.74 | 34.56 | 21.94 | 13.11 | 7.43 | 4.02 | 2.09 | 1.05 | 0.51 |
| 10 dice | 97.17 | 88.70 | 74.71 | 57.93 | 41.57 | 27.83 | 17.53 | 10.46 | 5.95 | 3.25 | 1.71 | 0.87 |
| 11 dice | 98.02 | 91.49 | 79.75 | 64.48 | 48.45 | 34.02 | 22.47 | 14.06 | 8.39 | 4.79 | 2.63 | 1.40 |
| 12 dice | 98.61 | 93.63 | 83.91 | 70.31 | 55.00 | 40.31 | 27.83 | 18.19 | 11.33 | 6.75 | 3.87 | 2.14 |
| 13 dice | 99.03 | 95.25 | 87.31 | 75.41 | 61.13 | 46.56 | 33.44 | 22.77 | 14.76 | 9.15 | 5.46 | 3.13 |
| 14 dice | 99.32 | 96.47 | 90.06 | 79.80 | 66.73 | 52.61 | 39.19 | 27.70 | 18.64 | 12.00 | 7.42 | 4.42 |
| 15 dice | 99.52 | 97.38 | 92.26 | 83.54 | 71.77 | 58.36 | 44.94 | 32.87 | 22.91 | 15.27 | 9.78 | 6.03 |
Rote actions
This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the rote-action system.
| Successes | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 die | 51.00 | 5.10 | 0.51 | 0.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 2 dice | 75.99 | 31.00 | 5.44 | 0.77 | 0.10 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 3 dice | 88.23 | 55.17 | 20.89 | 4.70 | 0.83 | 0.13 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 4 dice | 94.23 | 72.63 | 40.12 | 14.88 | 3.82 | 0.79 | 0.14 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 |
| 5 dice | 97.17 | 83.94 | 57.83 | 29.52 | 10.93 | 3.04 | 0.69 | 0.13 | 0.02 | 0.00 | 0.00 | 0.00 |
| 6 dice | 98.61 | 90.83 | 71.84 | 45.36 | 21.99 | 8.17 | 2.41 | 0.59 | 0.12 | 0.02 | 0.00 | 0.00 |
| 7 dice | 99.32 | 94.87 | 81.93 | 59.87 | 35.36 | 16.56 | 6.18 | 1.89 | 0.49 | 0.11 | 0.02 | 0.00 |
| 8 dice | 99.66 | 97.17 | 88.76 | 71.80 | 49.06 | 27.52 | 12.57 | 4.72 | 1.49 | 0.41 | 0.10 | 0.02 |
| 9 dice | 99.83 | 98.46 | 93.18 | 80.89 | 61.57 | 39.75 | 21.42 | 9.61 | 3.63 | 1.17 | 0.33 | 0.08 |
| 10 dice | 99.92 | 99.17 | 95.94 | 87.43 | 72.08 | 51.93 | 31.99 | 16.69 | 7.39 | 2.80 | 0.92 | 0.27 |
| 11 dice | 99.96 | 99.55 | 97.63 | 91.94 | 80.36 | 63.03 | 43.27 | 25.62 | 13.02 | 5.70 | 2.17 | 0.73 |
| 12 dice | 99.98 | 99.76 | 98.63 | 94.94 | 86.57 | 72.49 | 54.29 | 35.72 | 20.46 | 10.18 | 4.42 | 1.69 |
| 13 dice | 99.99 | 99.87 | 99.22 | 96.88 | 91.03 | 80.12 | 64.34 | 46.20 | 29.29 | 16.30 | 7.97 | 3.44 |
| 14 dice | 99.99 | 99.93 | 99.56 | 98.11 | 94.14 | 85.99 | 72.97 | 56.31 | 38.92 | 23.89 | 12.98 | 6.25 |
| 15 dice | 99.99 | 99.96 | 99.75 | 98.87 | 96.24 | 90.36 | 80.05 | 65.52 | 48.71 | 32.54 | 19.41 | 10.32 |