D6 Probabilities with the Wild Die
So, this problem keeps coming up again and again in various forums dealing with the loved and, for the most part, forgotten system from West End Games, D6 System. The problem is the presence of the "Wild Die" that was introduced in Star Wars 2nd Edition. As usual, there isn't a whole lot of grey space in this debate... or shall we more accurately say "Love/Hate".
I created a small perl script that takes in the number of iterations and the number of dice that are part of the roll and outputs the list of target numbers that were rolled, how many times they were rolled and what % of the rolls do they represent. The script takes into account the result of the Wild Die. I settled on 1,000,000 iterations as it seemed to give a consistent sample response when run several times.
The first graph is the Probability Distribution Function plotted for 1D - 12D (pips just shift the chart a step per pip to the right). 1D obviously has the greatest variation since it is just the wild die. By the time a person is rolling 5D the graphs have settled down to having a fairly normal looking bell curve, but with a long tail extending out to the far right from the "exploding" d6.
The second graph is the Cumulative Distribution Function plotted for the same 1D - 12D. The % is the chance to roll that value or higher with that many dice (with a small error as Excel didn't like putting the 0 data points on the X=0 spot). Again, the curves show a smooth falloff once the dice reaches 5D.
The graphs for the lack of a Wild Die are fairly simple and are scattered all over the Net for anyone interested in finding them.
For an interesting analysis of the math for Exploding Dice (not just a single Wild Die), check out this Blog entry A few people on the Weg Fan Forums asked for a copy of the table that was used to create some of the above graphs. Below is the table of data for the CDF, which, read another way, is the chance, when rolling a certain number of D6s (columns) to roll equal to or greater than a given value (rows). The table is a little squashed on this theme, but all the columns are present.
1D | 2D | 3D | 4D | 5D | 6D | 7D | 8D | 9D | 10D | 11D | 12D | |
0 | 100.00% | 100.00% | ||||||||||
1 | 100.00% | 100.00% | 100.00% | |||||||||
2 | 83.36% | 100.00% | 94.90% | 100.00% | ||||||||
3 | 66.68% | 83.24% | 90.74% | 98.80% | 100.00% | |||||||
4 | 49.98% | 80.46% | 87.47% | 96.70% | 99.73% | 100.00% | ||||||
5 | 33.31% | 74.92% | 84.70% | 94.08% | 99.02% | 99.94% | 100.00% | |||||
6 | 33.31% | 66.57% | 81.90% | 91.20% | 97.80% | 99.75% | 99.99% | 100.00% | ||||
7 | 16.68% | 55.50% | 78.66% | 88.29% | 96.09% | 99.32% | 99.94% | 100.00% | 100.00% | |||
8 | 13.90% | 44.39% | 74.05% | 85.43% | 94.00% | 98.57% | 99.81% | 99.99% | 100.00% | 100.00% | ||
9 | 11.12% | 32.81% | 67.56% | 82.42% | 91.60% | 97.47% | 99.54% | 99.95% | 100.00% | 100.00% | ||
10 | 8.32% | 23.57% | 59.14% | 78.88% | 88.99% | 96.00% | 99.08% | 99.87% | 99.99% | 100.00% | 100.00% | |
11 | 5.55% | 16.63% | 49.70% | 74.31% | 86.21% | 94.18% | 98.35% | 99.70% | 99.96% | 100.00% | 100.00% | 100.00% |
12 | 5.55% | 12.00% | 39.95% | 68.55% | 83.10% | 92.06% | 97.35% | 99.40% | 99.91% | 99.99% | 100.00% | 100.00% |
13 | 2.79% | 9.69% | 30.79% | 61.57% | 79.46% | 89.61% | 96.01% | 98.94% | 99.80% | 99.97% | 100.00% | 100.00% |
14 | 2.32% | 7.41% | 23.16% | 53.58% | 75.08% | 86.93% | 94.34% | 98.25% | 99.61% | 99.94% | 99.99% | 100.00% |
15 | 1.85% | 5.48% | 16.99% | 45.11% | 69.84% | 83.84% | 92.42% | 97.31% | 99.31% | 99.87% | 99.98% | 100.00% |
16 | 1.39% | 3.93% | 12.44% | 36.68% | 63.66% | 80.30% | 90.19% | 96.11% | 98.85% | 99.75% | 99.96% | 100.00% |
17 | 0.93% | 2.77% | 9.18% | 28.93% | 56.66% | 76.13% | 87.62% | 94.62% | 98.20% | 99.54% | 99.91% | 99.99% |
18 | 0.93% | 1.99% | 6.88% | 22.18% | 49.10% | 71.24% | 84.68% | 92.81% | 97.35% | 99.23% | 99.84% | 99.97% |
19 | 0.47% | 1.61% | 5.20% | 16.67% | 41.38% | 65.59% | 81.25% | 90.73% | 96.24% | 98.79% | 99.71% | 99.94% |
20 | 0.39% | 1.24% | 3.87% | 12.36% | 33.86% | 59.22% | 77.26% | 88.29% | 94.88% | 98.17% | 99.51% | 99.89% |
21 | 0.31% | 0.91% | 2.84% | 9.11% | 26.99% | 52.37% | 72.67% | 85.46% | 93.23% | 97.38% | 99.22% | 99.81% |
22 | 0.24% | 0.66% | 2.07% | 6.73% | 21.03% | 45.25% | 67.45% | 82.20% | 91.24% | 96.35% | 98.81% | 99.67% |
23 | 0.16% | 0.46% | 1.52% | 5.02% | 16.00% | 38.10% | 61.59% | 78.46% | 88.93% | 95.09% | 98.25% | 99.47% |
24 | 0.16% | 0.33% | 1.14% | 3.74% | 12.03% | 31.37% | 55.27% | 74.19% | 86.25% | 93.55% | 97.50% | 99.19% |
25 | 0.08% | 0.26% | 0.88% | 2.79% | 8.96% | 25.16% | 48.62% | 69.33% | 83.11% | 91.69% | 96.54% | 98.79% |
26 | 0.07% | 0.20% | 0.64% | 2.06% | 6.63% | 19.77% | 41.88% | 63.87% | 79.50% | 89.51% | 95.37% | 98.27% |
27 | 0.05% | 0.15% | 0.48% | 1.52% | 4.92% | 15.21% | 35.31% | 57.97% | 75.48% | 86.95% | 93.90% | 97.58% |
28 | 0.04% | 0.11% | 0.35% | 1.12% | 3.63% | 11.54% | 29.13% | 51.73% | 70.87% | 84.02% | 92.18% | 96.69% |
29 | 0.03% | 0.08% | 0.26% | 0.84% | 2.71% | 8.65% | 23.51% | 45.35% | 65.79% | 80.66% | 90.13% | 95.58% |
30 | 0.03% | 0.06% | 0.19% | 0.62% | 2.01% | 6.42% | 18.61% | 38.96% | 60.31% | 76.82% | 87.74% | 94.25% |
31 | 0.01% | 0.05% | 0.15% | 0.46% | 1.49% | 4.76% | 14.48% | 32.87% | 54.41% | 72.53% | 84.97% | 92.62% |
32 | 0.01% | 0.04% | 0.11% | 0.34% | 1.11% | 3.56% | 11.10% | 27.22% | 48.38% | 67.74% | 81.77% | 90.69% |
33 | 0.01% | 0.03% | 0.08% | 0.25% | 0.82% | 2.63% | 8.39% | 22.07% | 42.32% | 62.51% | 78.12% | 88.43% |
34 | 0.01% | 0.02% | 0.06% | 0.19% | 0.61% | 1.96% | 6.29% | 17.62% | 36.38% | 56.98% | 74.03% | 85.78% |
35 | 0.00% | 0.01% | 0.04% | 0.14% | 0.45% | 1.46% | 4.68% | 13.79% | 30.67% | 51.21% | 69.52% | 82.80% |
36 | 0.00% | 0.01% | 0.03% | 0.11% | 0.34% | 1.08% | 3.47% | 10.62% | 25.43% | 45.33% | 64.63% | 79.39% |
37 | 0.00% | 0.01% | 0.02% | 0.08% | 0.25% | 0.80% | 2.57% | 8.10% | 20.69% | 39.53% | 59.43% | 75.55% |
38 | 0.00% | 0.00% | 0.02% | 0.06% | 0.18% | 0.60% | 1.91% | 6.09% | 16.59% | 33.91% | 53.96% | 71.29% |
39 | 0.00% | 0.00% | 0.01% | 0.05% | 0.13% | 0.45% | 1.41% | 4.55% | 13.02% | 28.60% | 48.29% | 66.70% |
40 | 0.00% | 0.00% | 0.01% | 0.03% | 0.10% | 0.33% | 1.06% | 3.38% | 10.10% | 23.71% | 42.70% | 61.72% |
41 | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% | 0.25% | 0.78% | 2.53% | 7.75% | 19.31% | 37.16% | 56.52% |
42 | 0.00% | 0.00% | 0.01% | 0.02% | 0.05% | 0.19% | 0.58% | 1.87% | 5.90% | 15.52% | 31.86% | 51.10% |
43 | 0.00% | 0.00% | 0.00% | 0.01% | 0.04% | 0.13% | 0.42% | 1.39% | 4.44% | 12.27% | 26.86% | 45.61% |
44 | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% | 0.10% | 0.31% | 1.03% | 3.33% | 9.56% | 22.32% | 40.11% |
45 | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% | 0.24% | 0.77% | 2.48% | 7.36% | 18.29% | 34.83% |
46 | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.05% | 0.18% | 0.58% | 1.85% | 5.61% | 14.73% | 29.79% |
47 | 0.00% | 0.00% | 0.00% | 0.01% | 0.01% | 0.04% | 0.13% | 0.43% | 1.37% | 4.23% | 11.68% | 25.13% |
48 | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% | 0.10% | 0.32% | 1.01% | 3.16% | 9.17% | 20.90% |
49 | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% | 0.23% | 0.75% | 2.37% | 7.11% | 17.12% |
50 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.02% | 0.05% | 0.17% | 0.57% | 1.75% | 5.44% | 13.84% |
51 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.04% | 0.13% | 0.42% | 1.30% | 4.16% | 11.04% |
52 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% | 0.09% | 0.30% | 0.97% | 3.13% | 8.69% |
53 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% | 0.23% | 0.72% | 2.34% | 6.76% |
54 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.02% | 0.05% | 0.17% | 0.53% | 1.75% | 5.19% |
55 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.04% | 0.13% | 0.40% | 1.29% | 3.97% |
56 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% | 0.10% | 0.30% | 0.96% | 3.00% |
57 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% | 0.22% | 0.71% | 2.27% |
58 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.02% | 0.05% | 0.16% | 0.53% | 1.69% |
59 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.04% | 0.12% | 0.39% | 1.26% |
60 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% | 0.09% | 0.29% | 0.94% |
61 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% | 0.22% | 0.70% |
62 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.05% | 0.16% | 0.52% |
63 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.04% | 0.12% | 0.38% |
64 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% | 0.09% | 0.28% |
65 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% | 0.21% |
66 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.02% | 0.05% | 0.16% |
67 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.04% | 0.12% |
68 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% | 0.09% |
69 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.02% | 0.07% |
70 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.01% | 0.05% |
71 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% |
72 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.01% | 0.03% |