Fantastic Friday: Probability Distribution
One of areas which all role-playing games share (as opposed to, say, story games) is some amount of random chance affecting things. Some games have randomly-generated characteristics used to define a character. Even in games where characters aren't randomly-generated at all, dice are used in determining the results when a player attempts to overcome a challenge where the outcome (one way or the other) is not guaranteed.
For many years in the early days of the RPG hobby, most games used random numbers for character generation. Since there was a dearth of material for any one game (even the big dog in the early days, TSR, didn't originally produce scenarios -- the first scenario produced for D&D was by a company called Wee Warriors), many hobbyists converted material back and forth. This is simple to do if you're handy with math, and trivial now, when there are websites where the computer will do the work for you.
I was reading an article by Graeme Davis from Imagine #18 ("Games Without Frontiers," from September 1984), where Anthony P. Allan has worked out the probabilities for many different randomizers of the day. I love charts, so here we are. In all cases, n indicates the number of times that result comes up, p% indicates the likelihood of getting that result exactly, +P% indicates the likelihood of getting that result or higher, -P% indicates the likelihood of getting that result or lower. I found one error in going through the charts, which is fixed below.
2D6
Score | n | p% | +P% | -P% |
---|---|---|---|---|
2 | 1 | 2.778 | 100.000 | 2.778 |
3 | 2 | 5.556 | 97.222 | 8.333 |
4 | 3 | 8.333 | 91.667 | 16.667 |
5 | 4 | 11.111 | 83.333 | 27.778 |
6 | 5 | 13.889 | 72.222 | 41.667 |
7 | 6 | 16.667 | 58.333 | 58.333 |
8 | 5 | 13.889 | 41.667 | 72.222 |
9 | 4 | 11.111 | 27.778 | 83.333 |
10 | 3 | 8.333 | 16.667 | 91.667 |
11 | 2 | 5.556 | 8.333 | 97.222 |
12 | 1 | 2.778 | 2.778 | 100.000 |
3D6
Score | n | p% | +P% | -P% |
---|---|---|---|---|
3 | 1 | 0.463 | 100.000 | 0.463 |
4 | 3 | 1.389 | 99.537 | 1.852 |
5 | 6 | 2.778 | 98.148 | 4.630 |
6 | 10 | 4.630 | 95.370 | 9.259 |
7 | 15 | 6.944 | 90.741 | 16.204 |
8 | 21 | 9.722 | 83.796 | 25.926 |
9 | 25 | 11.574 | 74.074 | 37.500 |
10 | 27 | 12.500 | 62.500 | 50.000 |
11 | 27 | 12.500 | 50.000 | 62.500 |
12 | 25 | 11.574 | 37.500 | 74.074 |
13 | 21 | 9.722 | 25.926 | 83.796 |
14 | 15 | 6.944 | 16.204 | 90.741 |
15 | 10 | 4.630 | 9.259 | 95.370 |
16 | 6 | 2.778 | 4.630 | 98.148 |
17 | 3 | 1.389 | 1.852 | 99.537 |
18 | 1 | 0.463 | 0.463 | 100.000 |
4D6
Score | n | p% | +P% | -P% |
---|---|---|---|---|
4 | 1 | 0.077 | 100.000 | 0.077 |
5 | 4 | 0.309 | 99.923 | 0.386 |
6 | 10 | 0.772 | 99.614 | 1.157 |
7 | 20 | 1.543 | 98.843 | 2.701 |
8 | 35 | 2.701 | 97.299 | 5.401 |
9 | 56 | 4.321 | 94.599 | 9.722 |
10 | 80 | 6.173 | 90.278 | 15.895 |
11 | 104 | 8.025 | 84.105 | 23.920 |
12 | 125 | 9.645 | 76.080 | 33.565 |
13 | 140 | 10.802 | 66.435 | 44.367 |
14 | 146 | 11.265 | 55.633 | 55.633 |
15 | 140 | 10.802 | 44.367 | 66.435 |
16 | 125 | 9.645 | 33.565 | 76.080 |
17 | 104 | 8.025 | 23.920 | 84.105 |
18 | 80 | 6.173 | 15.895 | 90.278 |
19 | 56 | 4.321 | 9.722 | 94.599 |
20 | 35 | 2.701 | 5.401 | 97.299 |
21 | 20 | 1.543 | 2.701 | 98.843 |
22 | 10 | 0.772 | 1.157 | 99.614 |
23 | 4 | 0.309 | 0.386 | 99.923 |
24 | 1 | 0.077 | 0.077 | 100.000 |
4D6, dropping the lowest
Score | n | p% | +P% | -P% |
---|---|---|---|---|
3 | 1 | 0.077 | 100.000 | 0.077 |
4 | 4 | 0.309 | 99.923 | 0.386 |
5 | 10 | 0.772 | 99.614 | 1.157 |
6 | 21 | 1.620 | 98.843 | 2.778 |
7 | 38 | 2.932 | 97.222 | 5.710 |
8 | 62 | 4.784 | 94.290 | 10.494 |
9 | 91 | 7.022 | 89.506 | 17.515 |
10 | 122 | 9.414 | 82.485 | 26.929 |
11 | 148 | 11.420 | 73.071 | 38.349 |
12 | 167 | 12.886 | 61.651 | 51.235 |
13 | 172 | 13.272 | 48.765 | 64.506 |
14 | 160 | 12.346 | 35.494 | 76.852 |
15 | 131 | 10.108 | 23.148 | 86.960 |
16 | 94 | 7.253 | 13.040 | 94.213 |
17 | 54 | 4.167 | 5.787 | 98.380 |
18 | 21 | 1.620 | 1.620 | 100.000 |
2D10
Score | n | p% | +P% | -P% |
---|---|---|---|---|
2 | 1 | 1.000 | 100.000 | 1.000 |
3 | 2 | 2.000 | 99.000 | 3.000 |
4 | 3 | 3.000 | 97.000 | 6.000 |
5 | 4 | 4.000 | 94.000 | 10.000 |
6 | 5 | 5.000 | 90.000 | 15.000 |
7 | 6 | 6.000 | 85.000 | 21.000 |
8 | 7 | 7.000 | 79.000 | 28.000 |
9 | 8 | 8.000 | 72.000 | 36.000 |
10 | 9 | 9.000 | 64.000 | 45.000 |
11 | 10 | 10.000 | 55.000 | 55.000 |
12 | 9 | 9.000 | 45.000 | 64.000 |
13 | 8 | 8.000 | 36.000 | 72.000 |
14 | 7 | 7.000 | 28.000 | 79.000 |
15 | 6 | 6.000 | 21.000 | 85.000 |
16 | 5 | 5.000 | 15.000 | 90.000 |
17 | 4 | 4.000 | 10.000 | 94.000 |
18 | 3 | 3.000 | 6.000 | 97.000 |
19 | 2 | 2.000 | 3.000 | 99.000 |
20 | 1 | 1.000 | 1.000 | 100.000 |
4D5 (did anyone ever use this?)
Score | n | p% | +P% | -P% |
---|---|---|---|---|
4 | 1 | 0.160 | 100.000 | 0.160 |
5 | 4 | 0.640 | 99.840 | 0.800 |
6 | 10 | 1.600 | 99.200 | 2.400 |
7 | 20 | 3.200 | 97.600 | 5.600 |
8 | 35 | 5.600 | 94.400 | 11.200 |
9 | 52 | 8.320 | 88.800 | 19.520 |
10 | 68 | 10.880 | 80.480 | 30.400 |
11 | 80 | 12.800 | 43.200 | 69.600 |
12 | 85 | 13.600 | 56.800 | 56.800 |
13 | 80 | 12.800 | 43.200 | 69.600 |
14 | 68 | 10.880 | 30.400 | 80.480 |
15 | 52 | 8.320 | 19.520 | 88.800 |
16 | 35 | 5.600 | 11.200 | 94.400 |
17 | 20 | 3.200 | 5.600 | 97.600 |
18 | 10 | 1.600 | 2.400 | 99.200 |
19 | 4 | 0.640 | 0.800 | 99.840 |
20 | 1 | 0.160 | 0.160 | 100.000 |
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